Ideal Theory in Rings

arXiv:1401.2577v1 [math.RA] 11 Jan 2014

(Idealtheorie in Ringbereichen)
Emmy Noether
Translated by Daniel Berlyne
January 14, 2014

Contents.
§1.
§2.
§3.
§4.
§5.
§6.
§7.
§8.
§9.
§10.
§11.
§12.

Introduction.
Ring, ideal, finiteness condition.
Representation of an ideal as the least common multiple of finitely
many irreducible ideals.
Equality of the number of components in two different
decompositions into irreducible ideals.
Primary ideals. Uniqueness of the prime ideals belonging to two
different decompositions into irreducible ideals.
Representation of an ideal as the least common multiple of maximal
primary ideals. Uniqueness of the associated prime ideals.
Unique representation of an ideal as the least common multiple of
relatively prime irreducible ideals.
Uniqueness of the isolated ideals.
Unique representation of an ideal as the product of coprime
irreducible ideals.
Development of the study of modules. Equality of the number of
components in decompositions into irreducible modules.
Special case of the polynomial ring.
Examples from number theory and the theory of differential
expressions.
Example from elementary divisor theory.
Translator’s notes.
Acknowledgements.
1

As this paper was originally written in the early twentieth century, there
are a number of mathematical terms used that do not have an exact modern
equivalent, and as such may be ambiguous in meaning. Such terms are
underlined as they appear in the text, and an explanation is given at the end
of the paper to clarify their meanings.

Introduction.
This paper aims to convert the decomposition theorems for the integers
or the decomposition of ideals in algebraic number fields into theorems for
ideals in arbitrary integral domains (and rings in general). To understand
this correspondence, we consider the decomposition theorems for the integers
in a form somewhat different from the commonly given formulation.
In the equation
a “ p̺11 p̺22 . . . p̺σσ “ q1 q2 . . . qσ
take the prime powers qi to be components of the decomposition with the
following characteristic properties:
1. They are pairwise coprime ; and no q can be written as a product of
pairwise coprime numbers, so in this sense irreducibility holds.
2. Each two components qi and qk are relatively prime; that is to say,
if bqi is divisible by qk , then b is divisible by qk . Irreducibility also holds in
this sense.
3. Every q is primary; that is to say, if a product bc is divisible by q, but
b is not divisible by q, then a power1 of c is divisible by q. The representation
furthermore consists of maximal primary components, since the product of
two different q is no longer primary. The q are also irreducible in relation
to the decomposition into maximal primary components.
4. Each q is irreducible in the sense that it cannot be written as the
least common multiple of two proper divisors.
The connection between these primary numbers q and the prime numbers p is that for every q there is one and (disregarding sign) only one p that
is a divisor of q and a power of which is divisible by q: the associated prime
number. If p̺ is the lowest such power — ̺ being the exponent from q —
then in particular, p̺ is equal to q here. The uniqueness theorem can now
1

If this power is always the first, then as is well-known it concerns a prime number.

2

be stated as follows:
Given two different decompositions of an integer into irreducible maximal primary components q, each decomposition has the same number of
components, the same associated prime numbers (up to sign) and the same
exponents. Because p̺ “ q, it also follows that the q themselves are the same
(up to sign).
As is well-known, the uncertainty brought about by the sign is eradicated
if instead of numbers, the ideals derived from them (all numbers divisible
by a) are considered; the formulation then holds exactly for the unique
decomposition of ideals of (finite) algebraic number fields into powers of
prime ideals.
In the following (§1) a general ring is considered, which must only satisfy
the finiteness condition that every ideal of the domain has a finite ideal basis.
Without such a finiteness condition, irreducible and prime ideals need not
exist, as shown by the ring of all algebraic integers, in which there is no
decomposition into prime ideals.
It is shown that — corresponding to the four characteristic properties
of the component q — in general four separate decompositions exist, which
follow successively from each other through subdivision. Thereby the decomposition into coprime irreducible ideals behaves as a factorisation, and the
remaining three decompositions behave as a reduced (§2) representation as
the least common multiple. Also, the connection between a primary ideal —
the irreducible ideals are also primary — and the corresponding prime ideal
is preserved: every primary ideal Q uniquely determines a corresponding
prime ideal P which is a divisor of Q and a power of which is divisible by
Q. If P̺ is the lowest such power — ̺ being the exponent of Q — then P̺
does not need to coincide with Q here, however. The uniqueness theorem
can now be expressed as follows:
The decompositions 1 and 2 are unique; given two different decompositions 3 or 4, the number of components and the corresponding prime ideals are the same;2 the isolated ideals (§7) occurring in the components are
uniquely determined.
For the proofs of the decomposition theorems, the notable “Theorem of
the Finite Chain” for finite modules, first by Dedekind, is followed using the
2

Moreover, the exponents are presumably also the same, and more generally the corresponding components are isomorphic.

3

finiteness condition, and from this we deduce representation 4 of each ideal
as the least common multiple of finitely many irreducible ideals. By revising
the statement of the reducibility of a component, the fundamental uniqueness theorem for decomposition 4 into irreducible ideals is produced. By
concluding that each of the remaining decompositions are given by finitely
many components, the uniqueness theorems for these emerge as a result of
uniqueness theorem 4.
Finally it is shown (§9) that the representation through finitely many
irreducible components also holds under weaker requirements; the commutativity of the ring is not necessary, and it suffices to consider a module
in relation to the ring instead of an ideal. In this more general case the
equality of the number of components for two different decompositions still
holds, while the notions of prime and primary are restricted to commutativity and the concept of an ideal; in contrast, the notion of coprime for ideals
is retained in non-commutative rings.
The simplest ring for which the four separate decompositions actually
occur is the ring of all polynomials in n variables with arbitrary complex
coefficients. The particular decompositions can be deduced here to be irrational due to the properties of algebraic figures, and the uniqueness theorem for the corresponding prime ideals is equivalent to the Fundamental
Theorem of Elimination Theory regarding the unique decomposability of algebraic figures into irreducible elements. Further examples are given by all
finite integral domains of polynomials (§10). In fact, the simple ring of all
even numbers, or more generally all numbers divisible by a given number, is
also an example for the separate decompositions (§11). An example of ideal
theory in non-commutative rings is provided by elementary divisor theory
(§12), where unique decomposition into irreducible ideals, or classes, holds.
These irreducible classes characterise completely the irreducible parts of elementary divisors, and in rings where the usual elementary divisor theory
breaks down can perhaps be considered as their equivalent.
In the available literature the following are to be noted: the decomposition into maximal primary ideals is given by Lasker for the polynomial
ring with arbitrary complex or integer coefficients, and taken further by
Macaulay at particular points.3 Both concern themselves with elimination
3

E. Lasker, Zur Theorie der Moduln und Ideale. Math. Ann. 60 (1905), p20, Theorems
VII and XIII. — F. S. Macaulay, On the Resolution of a given Modular System into
Primary Systems including some Properties of Hilbert Numbers. Math. Ann. 74 (1913),

4

theory, therefore using the fact that a polynomial can be expressed uniquely
as the product of irreducible polynomials. In fact, the decomposition theorems for ideals are independent of this hypothesis, as ideal theory in algebraic number fields allows one to suppose and as this paper shows. The
primary ideal is also defined by Lasker and Macaulay using concepts from
elimination theory.
The decomposition into irreducible ideals and into relatively prime irreducible ideals appears also not to be remarked upon for the polynomial ring
in the available literature; only a remark by Macaulay on the uniqueness of
the isolated primary ideals can be found.
The decomposition into coprime irreducible ideals is given by Schmeidler4 for the polynomial ring, using elimination theory for the proof of finiteness. However, here the uniqueness theorem is stated only for classes of ideals, not for the ideals themselves. This last uniqueness theorem can be found
in a joint paper,5 where it refers to ideals in non-commutative polynomial
rings. Only the finite ideal basis is made use of here, so theorems and methods for general rings remain to be addressed, becoming more of a problem
through this paper in terms of equality in size (§11). The present researches
give a strong generalisation and further development of the underlying concepts of both of these works. The basis of both works is the transition from
the expression as the least common multiple to an additive decomposition
of the system of residue classes. Here the least common multiple is kept for
the sake of a simpler representation; the additive decomposition then corresponds to the conversion of the idea of reducibility into a property of the
complement (§3). Therefore all given theorems are left to be reflected upon
and understood in the form of additive decomposition theorems for the system of residue classes and known subsystems, which is essentially equivalent
to the reflections of the joint paper. This system of residue classes forms a
ring of the same generality as originally laid down; that is, every ring can
be regarded as a system of residue classes of the ideal which corresponds to
the collection of all identity relations between the elements of the ring; or
also a subsystem of these relations, by assuming the remaining relations are
p66.
4
W. Schmeidler, Über Moduln und Gruppen hyperkomplexer Größen. Math. Zeitschr.
3 (1919), p29.
5
E. Noether - W. Schmeidler, Moduln in nichtkommutativen Bereichen, insbesondere
aus Differential- und Differenzenausdrücken. Math. Zeitschr. 8 (1920), p1.

5

also satisfied in the ring.
This remark also gives the classification in Fraenkel’s papers.6 Fraenkel
considers additive decompositions of rings that depend on such restrictive
conditions (existence of regular elements, division by these, decomposability
requirement), which coincide with the four decompositions for the corresponding ideal. Because of this concurrence, its finiteness condition also
means that the ideal only has finitely many proper divisors — apart from
some exceptional cases — a restriction no stricter than ours. Fraenkel’s
starting point is different, dependent on the essentially algebraic goals of his
work; through algebraic extension he achieved more general rings with fewer
restricting conditions.

§1. Ring, ideal, finiteness requirement.
1. Let Σ be a (commutative) ring in an abstract definition;7 that is
to say, Σ consists of a system of elements a, b, c, . . . , f, g, h, . . . in which a
relation satisfying the usual requirements is defined as equality, and in which
each two ring elements a and b combine uniquely through two operations,
addition and multiplication, to give a third element, given by the sum a ` b
and the product a¨b. The ring and the otherwise entirely arbitrary operations
must satisfy the following laws:
1. The associative law of addition: pa ` bq ` c “ a ` pb ` cq.
2. The commutative law of addition: a ` b “ b ` a.
3. The associative law of multiplication: pa ¨ bq ¨ c “ a ¨ pb ¨ cq.
4. The commutative law of multiplication: a ¨ b “ b ¨ a.
5. The distributive law: a ¨ pb ` cq “ a ¨ b ` a ¨ c.
6. The law of unrestricted and unique subtraction.
In Σ there is a single element x which satisfies the equation a ` x “ b.
(Written x “ b ´ a.)
6

A. Fraenkel, Über die Teiler der Null und die Zerlegung von Ringen. J. f. M. 145
(1914), p139. Über gewisse Teilbereiche und Erweiterungen von Ringen. Professorial
dissertation, Leipzig, Teubner, 1916. Über einfache Erweiterungen zerlegbarer Ringe. J.
f. M. 151 (1920), p121.
7
The definition is taken from Fraenkel’s professorial dissertation, with omission of the
more restrictive requirements 6, I and II; instead the commutative law of addition must
be incorporated. It therefore concerns the laws defining a field, with the omission of the
multiplicative inverse.

6

The existence of the zero element follows from these properties; however,
a ring is not required to possess a unit, and the product of two elements can
be zero without either of the factors being zero. Rings for which a product
being zero implies one of the factors is zero, and which in addition possess a
unit, are called integral domains. For the finite sum a` a` ¨ ¨ ¨` a we use the
usual abbreviated notation na, where the integers n are solely considered as
shortened notation, not as ring elements, and are defined recursively through
a “ 1 ¨ a, na ` a “ pn ` 1qa.
2. Let an ideal M8 in Σ be understood to be a system of elements of Σ
such that the following two conditions are satisfied:
1. If M contains f , then M also contains a ¨ f , where a is an arbitrary
element of Σ.
2. If M contains f and g, then M also contains the difference f ´ g;
so if M contains f , then M also contains nf for all integers n.
If f is an element of M, then we write f ” 0 pMq, as is usual; and we say
that f is divisible by M. If every element of N is equal to some element of M,
and so divisible by M, then we say N is divsible by M, written N ” 0 pMq.
M is called a proper divisor of N if it contains elements not in N, and so is
not conversely divisible by N. If N ” 0 pMq and M ” 0 pNq, then N “ M.
The remaining familiar notions also remain valid, word for word. By the
greatest common divisor D “ pA, Bq of two ideals A and B, we understand
this to be all elements which are expressible in the form a ` b, where a is an
element of A and b is an element of B; D is also itself an ideal. Likewise, the
greatest common divisor D “ pA1 , A2 , . . . , Aν , . . . q of infinitely many ideals
is defined as all elements d which are expressible as the sum of the elements
of each of finitely many ideals: d “ ai1 ` ai2 ` ¨ ¨ ¨ ` ain ; here D is again itself
an ideal.
Should the ideal M contain in particular a finite number of elements
f1 , f2 , . . . , f̺ such that
M “ pf1 , . . . , f̺ q;
that is,
f “ a1 f1 ` ¨ ¨ ¨ ` a̺ f̺ ` n1 f1 ` . . . n̺ f̺ for all f ” 0 pMq,
8

Ideals are denoted by capital letters. The use of M brings to mind the example of the
ideals composed of polynomials usually called “modules”. Incidentally, §§1-3 use only the
properties of modules and not the properties of ideals; compare §9 as well.

7

where the ai are elements of the ring, the ni are integers, and so M forms a
finite ideal; f1 , . . . , f̺ forms an ideal basis.
In the following we now consider solely rings Σ which satisfy the finiteness condition: every ideal in Σ is finite, and so has an ideal basis.
3. The following underlying ideas all follow directly from the finiteness
condition:
Theorem I (Theorem of the Finite Chain):9 If M, M1 , M2 , . . . , Mν , . . .
is a countably infinite system of ideals in Σ in which each ideal is divisible
by the following one, then all ideals after a finite index n are identical;
Mn “ Mn`1 “ . . . . In other words: If M, M1 , M2 , . . . , Mν , . . . gives a
simply ordered chain of ideals such that each ideal is a proper divisor of its
immediate predecessor, then the chain terminates in a finite number of steps.
In particular, let D “ pM1 , M2 , . . . , Mν , . . . q be the greatest common
divisor of the system, and let f1 . . . fk be an always existing basis of D
resulting from the finiteness condition. Then it follows from the divisibility
assumption that every element of D is equal to an element of an ideal in the
chain; then it follows from
f “ g ` h,

g ” 0 pMr q,

h ” 0 pMs q,

pr ď sq

that g ” 0 pMs q and so f ” 0 pMs q. The corresponding statement holds if
f is the sum of several components. There is therefore also a finite index n
such that
f1 ” 0 pMn q; . . . ; fk ” 0 pMn q; D “ pf1 , . . . , fk q ” 0 pMn q.
Because conversely Mn ” 0 pDq, it follows that Mn “ D; and because
furthermore
Mn`i ” 0 pDq;

D “ Mn ” 0 pMn`i q,

it also follows that Mn`i “ D “ Mn for all i, whereupon the theorem is
proved.
9

Initially stated for modules by Dedekind: Zahlentheorie, Suppl. XI, §172, Theorem
VIII (4th condition); our proof and the term ”chain” is taken from there. For ideals of
polynomials by Lasker, loc. cit. p56 (lemma). In both cases the theorem finds only
specific applications, however. Our applications depend without exception on the axiom
of choice.

8

Note that conversely the existence of the ideal basis follows from this
theorem, so the finiteness condition could also have been stated in this basisfree form.

§2. Representation of an ideal as the least common
multiple of finitely many irreducible ideals.
Let the least common multiple rB1 , B2 , . . . , Bk s of the ideals B1 , B2 ,
. . . , Bk be defined as usual as the collection of elements which are divisible
by each of B1 , B2 , . . . , Bk , written as:
f ” 0 pBi q, pi “ 1, 2, . . . , kq implies f ” 0 prB1 , B2 , . . . , Bk sq
and vice versa. The least common multiple is itself an ideal; we call the
ideals Bi the components of the decomposition.
Definition I. A representation M “ rB1 , . . . , Bk s is called a reduced
representation if no Bi appears in the least common multiple Ai of the
remaining ideals, and if no Bi can be replaced with a proper divisor.10
If the conditions are only satisfied for the ideal Bi , then the representation is called reduced with respect to Bi . The least common multiple
Ai “ rB1 , . . . , Bi´1 , Bi`1 , . . . , Bk s is called the complement of Bi . Representations in which only the first condition is satisfied are called shortest
representations.
It suffices now, when considering the representation of an ideal as a lowest common multiple, to restrict ourselves to reduced representations, due
to the following lemma:
Lemma I. Every representation of an ideal as the least common multiple
of finitely many ideals can be replaced in at least one way by a reduced rep10

An example of a non-reduced representation is:
px2 , xyq “ rpxq, px2 , xy, y λ qs

for all exponents λ ě 2; the case λ “ 1, corresponding to the representation rpxq, px2 , yqs,
gives a reduced representation. (K. Hentzelt, who was killed in the war, gave this representation to me as the simplest example of a non-unique decomposition into primary
ideals.)

9

resentation; in particular, one such representation can be obtained through
successive decomposition.
Let M “ rB˚1 , . . . , B˚l s be an arbitrary representation of M. We can then
omit those B˚i which go into the least common multiple of the remaining
ideals. Because the remaining ideals still give M, the resulting representation
M “ rB1 , . . . , Bk s “ rAi , Bi s
satisfies the first condition, and so this is a shortest representation; and
this condition remains satisfied if some Bi is replaced with a proper divisor.
But the second condition is always satisfiable, by the Theorem of the Finite
Chain (Theorem I). For suppose
p1q

pνq

M “ rAi , Bi s “ rAi , Bi s “ ¨ ¨ ¨ “ rAi , Bi s, . . . ,
pνq

where each Bi is a proper divisor of its immediate predecessor, so that
p1q
pνq
the chain Bi , Bi , . . . , Bi , . . . must terminate in finitely many steps; in
pnq
pnq
the representation M “ rAi , Bi s, Bi can therefore not be replaced with
a proper divisor, and this holds a fortiori if Ai is replaced with a proper
divisor. Therefore if the algorithm is applied to each Bi by at each stage
using the complement together with the already reduced B, then a reduced
representation is formed.11
In order to obtain such a representation successively, it must be shown
that it follows from the individual reduced representations
M “ rB1 , C1 s, C1 “ rB2 , C2 s, . . . , Cl´1 “ rBl , Cl s
that the representation M “ rB1 , . . . , Bl , Cl s resulting from these is also
reduced. For this it is sufficient to show that if the representations M “
rB1 , . . . , B̺ , Cs and C “ rC1 , C2 s are reduced, then M “ rB1 , . . . , B̺ , C1 , C2 s
is also reduced. Indeed, by assumption no B appears in its complement;
were this the case for a Ci , then in the first representation, C would be replaceable with a proper divisor, contrary to the assumption, because due
to the second reduced representation, C1 and C2 are proper divisors of C;
the representation is therefore a shortest representation. Furthermore, by
11

The previous example shows that the given representation does not uniquely define
such a reduced representation. For px2 , xyq “ rpxq, px2 , xy, y λ qs, where λ ě 2, both
rpxq, px2 , yqs and rpxq, px2 , µx ` yqs for arbitrary µ are reduced representations.

10

assumption no B can be replaced with a proper divisor; were this the case
for a Ci , then this would correspond to the substitution of C with a proper
divisor, in contradiction with the assumption, because the representation
for C is reduced. The lemma is thus proved.
Definition II. An ideal M is called reducible if it can be expressed as
the least common multiple of two proper divisors; otherwise M is called irreducible.
We now prove the following via the Theorem of the Finite Chain (Theorem I) and using the reduced representation:
Theorem II. Every ideal can be expressed as the least common multiple
of finitely many irreducible ideals.12
An arbitrary ideal M is either irreducible, in which case M “ rMs is a
representation in the form required by Theorem II, or it is M “ rB1 , C1 s,
where B1 , C1 are proper divisors of M, and by Lemma I the representation
can be assumed to be reduced. The same choice is true of C1 ; either it is
irreducible or it has a reduced representation C1 “ rB2 , C2 s.
Continuing in this way, the following series of reduced representations is
obtained:
M “ rB1 , C1 s; C1 “ rB2 , C2 s; . . . ; Cν´1 “ rBν , Cν s; . . .

.

(1)

In the chain C1 , C2 , . . . , Cν , . . . , each Ci is a proper divisor of its immediate predecessor, and therefore the chain terminates in finitely many steps;
there is an index n such that Cn is irreducible. Furthermore, by Lemma
I the representation M “ rB1 , . . . , Bn , Cn s is reduced; Cn can therefore
not go into its complement An , and Cn cannot be replaced by any proper
divisor in the representation M “ rAn , Cn s. Replacing An with a proper
divisor13 if necessary so that the representation is reduced, it has therefore
12

The previous example shows that such a representation is in general not unique:
px , xyq “ rpxq, px2 , µx `yqs. Both components are irreducible for arbitrary µ. All divisors
of pxq are of the form px, gpyqq, where gpyq denotes a polynomial in y; therefore if the least
common multiple of two divisors also has this form, then it is a proper divisor of pxq.
px2 , µx ` yq has only one divisor px, yq, and so is necessarily also irreducible.
13
Indeed, the representation is also reduced with respect to An , as will be shown in §3
(Lemma IV) as the converse of Lemma I.
2

11

been shown that every reducible ideal admits a reduced representation as the
least common multiple of an irreducible ideal and an ideal complementary to
it. All Bi in series (1) can therefore without loss of generality be assumed
to be irreducible. Iteration of the above argument gives the existence of an
irreducible Cn , which proves Theorem II.

§3. Equality of the number of components in two
different decompositions into irreducible ideals.
To prove the equality of the number of components, it is first necessary
to express the reducibility and irreducibility of an ideal through properties
of its complement, via
Theorem III.14 Let the shortest representation M “ rA, Cs be reduced
with respect to C. Then a necessary and sufficient condition for C to be
reducible is the existence of two ideals N1 and N2 which are proper divisors
of M such that
N1 ” 0 pAq;

N2 ” 0 pAq;

rN1 , N2 s “ M.

(2)

From this also follows: If the conditions (2) are satisfied, and C is irreducible, then at least one of the Ni is not a proper divisor of M; Ni “ M.
Let C “ rC1 , C2 s, where C1 , C2 are proper divisors of C. Then it follows
that
M “ rA, Cs “ rA, C1 , C2 s “ rrA, C1 s, rA, C2 ss.
Here the ideals rA, Ci s are proper divisors of M, because otherwise rA, Cs
would not be reduced with respect to C. Because the divisibility by M is
also satisfied, condition (2) is proved to be necessary. (The representation
(2) is not reduced, because a rA, Ci s can be replaced with Ci .)
Conversely, now suppose (2) holds. We construct the ideals:
C1 “ pC, N1 q;

C2 “ pC, N2 q;

14

C˚ “ rC1 , C2 s.

Theorem III corresponds to the transition from modules to quotient groups in the
works of Schmeidler and Noether-Schmeidler (cf. the introduction). Here A corresponds
to the quotient group, and N1 and N2 to the subgroups into which the quotient group is
decomposed.

12

Then C is divisible by both C1 and C2 , and therefore also by the least common
multiple C˚ . In order to show the divisibility of C˚ by C, let f ” 0 pC˚ q;
therefore f ” 0 pC1 q and f ” 0 pC2 q, or also f “ c`n1 and f “ c̄`n2 , where
c, c̄, n1 , n2 are elements of C, N1 , N2 , and so in particular ni is divisible by
A. Therefore the difference
g “ c ´ c̄ “ n2 ´ n1
is divisible both by C and by A, and so by M. Because n1 “ n2 ` m, it
further holds that n1 (likewise n2 ) is divisible both by N1 and by N2 , and
so by M. Therefore
f “ c ` m;

f ” 0 pCq;

C˚ “ C.

Here C1 and C2 are proper divisors of C, since if Ci “ pC, Ni q “ C, then Ni
would be divisible by C, and so because of the divisibility by A, Ni would
be equal to M, in contradiction with the assumption. Hence C “ rC1 , C2 s is
identified as reducible; Theorem III is proved.
Note that almost identical reasoning also shows the following:
Lemma II. If the ideal C in a shortest representation M “ rA, Cs can
be replaced with a proper divisor, then C is reducible.
Let
M “ rA, Cs “ rA, C1 s,
and set
C˚ “ rC1 , pA, Cqs.
Then C is again divisible by C˚ . Furthermore, it follows from f ” 0 pC˚ q
that
f “ c1 “ a ` c.
The difference a “ c1 ´ c is therefore divisible both by A and by C1 , and
therefore by M; therefore c1 “ c ` m; f ” 0 pCq; C˚ “ C.
Because both C1 and pA, Cq are proper divisors of C by assumption,
C “ C˚ is thus identified as reducible.
13

An irreducible C can therefore not be replaced with a proper divisor.
Now let the following be two different shortest representations of M as
the least common multiple of (finitely many) irreducible ideals:
M “ rB1 , . . . , Bk s “ rD1 , . . . , Dl s.
These representations are both reduced according to the remark following
Lemma II. Now we shall first prove
Lemma III. For every complement Ai “ rB1 . . . Bi´1 Bi`1 . . . Bk s there
exists an ideal Dj such that M “ rAi , Dj s.
Set M “ rD1 , C1 s, C1 “ rD2 , C12 s and so forth, so that
M “ rAi , Ms “ rAi , D1 , C1 s “ rrAi D1 s, rAi C1 ss.
Here the conditions (2) from Theorem III are satisfied for N1 “ rAi , D1 s, N2 “
rAi , C1 s, because M “ rAi , Bi s is reduced with respect to Bi and the representation is a shortest one. Because Bi was assumed to be irreducible, one
Ni must necessarily be equal to M.
For N1 “ M the lemma would be proved; for N2 “ M it follows respectively that M “ rrAi D2 s, rAi C12 ss, where, by the same result, one component must again be equal to M. Continuing in this way, it follows either that
M “ rAi , Dj s, where j ă l, or that M “ rAi , C1...l´1 s; because C1...l´1 “ Dl ,
the lemma is thus proven.
Now, as a result of this, we have
Theorem IV. For two different shortest representations of an ideal as
the least common multiple of irreducible ideals, the number of components
is the same.
It follows from the lemma for the particular case i “ 1:
M “ rA1 , B1 s “ rA1 , Dj1 s “ rDj1 , B2 , . . . , Bk s.
Now consider both of the decompositions
M “ rDj1 , B2 , . . . , Bk s “ rD1 , D2 , . . . , Dl s,

14

and recall the earlier result with reference to the complement Ā2 “
rDj1 , B3 , . . . , Bk s of B2 . It then follows that
M “ rĀ2 , B2 s “ rĀ2 , Dj1 s “ rDj1 , Dj2 , B3 , . . . , Bk s;
and by extension of this procedure:
M “ rDj1 , Dj2 , . . . , Djk s.
Because now the representation M “ rD1 , . . . , Dl s is a shortest one by assumption, and so no D can be omitted, the different ones among the Dji
must exhaust all of the D; therefore it follows that k ě l. Should we swap
the B with the D throughout the lemma and the subsequent results, it then
follows accordingly that l ě k, and therefore that k “ l, which proves the
equality of the number of components. As a result of this it follows that the
ideals Dji are all different from each other, because otherwise there would
be fewer than k components in a shortest representation using the Dji ; the
notation can therefore be chosen so that Dji “ Di . By the same reasoning,
all intermediate representations M “ rDj1 , . . . , Dji , Bi`1 , . . . , Bk s are also
shortest and so, by the remark following Lemma II, reduced.
The equality of the number of components leads us to a converse of
Lemma I through
Lemma IV. If the components in a reduced representation are collected
into groups and the least common multiples of them constructed, then the
resulting representation is reduced. In other words: Given a reduced representation
M “ rC11 , . . . , C1µ1 ; . . . ; Cσ1 , . . . , Cσµσ s,
it follows that M “ rN1 , . . . , Nσ s “ rNi , Li s is also reduced, where Ni “
rCi1 , . . . , Ciµi s.
Firstly, note that Ni cannot go into its complement Li , since this is not
the case for any of its divisors Cij ; the representation is therefore a shortest
one. In order to show that Ni cannot be replaced with any proper divisor,
we split the C into their irreducible ideals B,15 so that the (by Lemma I)
15

By this we understand the B to always be shortest, and therefore reduced, representations.

15

reduced representations arise:
M “ rB11 , . . . , B1λ1 ; . . . ; Bσ1 , . . . , Bσλσ s;

Ni “ rBi1 , . . . , Biλi s.

Now let M “ rN˚i , Li s be reduced with respect to N˚i , where N˚i is a proper
divisor of Ni . Then by Lemma II:
Ni “ rN˚i , pNi , Li qs;
and this representation is necessarily reduced with respect to N˚i , because
otherwise N˚i can also be replaced with a proper divisor in M. Now also
replace pNi , Li q with a proper divisor where appropriate, so that a reduced
representation for Ni is achieved. If both components of Ni are now decomposed into irreducible ideals, then the number λi of irreducible ideals in Ni is
composed additively of those of the components; the number of irreducible
ideals corresponding to the proper divisor N˚i is therefore necessarily smaller
than λi . Then, however, the decomposition of M “ rN˚i , Ls into irreducible
ř
ideals leads to fewer than i λi ideals, in contradiction with the equality of
number of components. In the special case σ “ 2, it also follows that the
representation M “ rNi , Li s is reduced with respect to the complement Li .

§4. Primary ideals. Uniqueness of the prime ideals belonging to two different decompositions into
irreducible ideals.
The following concerns the connection between primary and irreducible
ideals.
Definition III. An ideal Q is called primary if a ¨ b ” 0 pQq, a ı 0 pQq
implies bx ” 0 pQq, where the exponent x is a finite number.
The definition can also be restated as follows: if a product a¨b is divisible
by Q, then either one factor is divisible by Q or a power of the other is. If
in particular x is always equal to 1, then the ideal is called a prime ideal.
From the definition of a primary (respectively prime) ideal follows, by
virtue of the existence of a basis, the definition using only products of ide-

16

als:16
Definition IIIa. An ideal Q is called primary if A ¨ B ” 0 pQq,
A ı 0 pQq necessarily implies Bλ ” 0 pQq. If λ is always equal to 1,
then the ideal is called a prime ideal. For a prime ideal P, it therefore always follows from A ¨ B ” 0 pPq, A ı 0 pPq that B ” 0 pPq.
Because Definition III is contained in IIIa for A “ paq, B “ pbq as a
special case, every ideal which is primary by IIIa is also primary by III.
Conversely, suppose Q is primary by III, and let the assumption of IIIa be
satisfied: A ¨ B ” 0 pQq, so that it therefore follows either that A ” 0 pQq,
or alternatively that there is at least one element a ” 0 pAq such that
a ¨ B ” 0 pQq and a ı 0 pQq. If now b1 , . . . , br is an ideal basis of B, then
by Definition III, since a ¨ bi ” 0 pQq, the following holds:
bx1 1 ” 0 pQq; . . . ; bxr r ” 0 pQq.
Because b “ f1 b1 `¨ ¨ ¨`fr br `n1 b1 `¨ ¨ ¨`nr br , for λ “ x1 `¨ ¨ ¨`xr the product
of λ elements b is therefore divisible by Q, which proves the fulfilment of
Definition IIIa for ideals primary by III. In particular, for prime ideals P
it follows from a ¨ B ” 0 pPq, and therefore also a ¨ b ” 0 pPq for every
b ” 0 pBq and a ı 0 pPq, that b ” 0 pPq and thus B ” 0 pPq. We have
therefore shown the two definitions to be equivalent.
The connection between primary and prime ideals shall be established
through the remark that the collection P of all elements p with the property
that a power of p is divisible by Q forms a prime ideal. It is immediately clear
that P is an ideal; because if the given property holds for p1 and p2 , it also
holds for ap1 and p1 ´ p2 . Furthermore, by the inference used in Definition
IIIa regarding the basis, there exists a number λ such that Pλ ” 0 pQq.
Now let
a ¨ b ” 0 pPq;

a ı 0 pPq,

so that it follows from the definition of P that
aλ ¨ bλ ” 0 pQq;
16

aλ ı 0 pQq;

The product A ¨ B of two ideals is understood to mean, as usual, the ideal consisting
of the collection of elements a ¨ b and their finite sums.

17

therefore by the definition of Q:
bλx ” 0 pQq

and hence

b ” 0 pPq,

which proves P to be a prime ideal. P is also defined as the greatest common
divisor of all ideals B with the property that a power of B is divisible by
Q. Every such B is by definition divisible by P; thus the greatest common
divisor D of these B is too. Conversely, P is itself one of the ideals B, so
is divisible by D, which proves that P “ D. P is therefore a prime ideal
which is a divisor of Q and a power of which is divisible by Q. It is uniquely
defined by these properties, because it follows from
Q ” 0 pPq;

Pλ ” 0 pQq;

Q ” 0 pP̄q;

Pλ ” 0 pP̄q;

P̄µ ” 0 pPq;

P̄µ ” 0 pQq

that

and so, by the properties of prime ideals,
P ” 0 pP̄q;

P̄ ” 0 pPq;

P “ P̄.

In conclusion, we have
Theorem V. For every primary ideal Q there exists one, and only one,
prime ideal P which is a divisor of Q and a power of which is divisible by
Q; P shall be referred to as the “associated prime ideal”.17 P is defined as
the greatest common divisor of all ideals B with the property that a power
of B is divisible by Q. If ̺ is the smallest number such that P̺ ” 0 pQq,
then ̺ shall be referred to as the exponent of Q.18
We now prove, as the connection between primary and irreducible:
17

The example M “ px2 , xyq shows that the converse does not hold. The prime ideal
pxq satisfies all requirements, but M is not primary.
18
It is not in general true that P̺ “ Q, as it is in the rings of the integers and the
algebraic integers; for example:
Q “ px2 , yq;

P “ px, yq;

P2 “ px2 , xy, y 2 q ” 0 pQq; but Q ı 0 pP2 q;

therefore Q differs from P2 .

18

Theorem VI. Every non-primary ideal is reducible; in other words, every irreducible ideal is primary.19
Let K be a non-primary ideal, so that by Definition III there exists at
least one pair of elements a, b such that
a ¨ b ” 0 pKq;

a ı 0 pKq;

bx ı 0 pKq for every x.

(3)

We now construct the two ideals
L0 “ pK, aq;

N0 “ pK, bq,

which by (3) are proper divisors of K, and by (3) it holds that
L0 ¨ N0 ” 0 pKq.

(4)

For the elements f of the least common multiple K0 “ rL0 , N0 s, the following
options now arise:
Either from
f ” 0 pL0 q;

f ” 0 pN0 q, that is f ” a1 ¨ b pKq

always follows a representation
f ” l0 ¨ b pKq;

l0 ” 0 pL0 q;

therefore, by (4), f ” 0 pKq, which implies K0 ” 0 pKq, and because K ”
0 pK0 q, it also holds that K “ K0 , by which K is proven to be reducible.
Or there is at least one f ” 0 pK0 q for which there exists no such l0 .
Using the a1 belonging to this f we then construct:
L1 “ pL0 , a1 q “ pK, a, a1 q;
19

N1 “ pK, b2 q.

The following example shows that the converse does not hold here:
Q “ px2 , xy, y λ q “ rpx2 , yq, px, y λ qs,

where λ ě 2. Here Q is primary, but reducible. (Q is primary because it contains all
products of powers of x and y with a total dimension of λ; for every polynomial without
a constant term, one power is therefore divisible by Q. However, should the polynomial
b in a ¨ b ” 0 pQq contain a constant term – so bx ı 0 pQq for every x – then a must be
divisible by Q, because every homogeneous component of a ¨ b is divisible by Q due to the
homogeneity of the basis polynomials of Q.)

19

Then, because a1 ¨ b ” 0 pL0 q by (4), it also holds that
L1 ¨ N1 ” 0 pKq;

(4’)

and L1 is a proper divisor of L0 .
For the elements f of K1 “ rL1 , N1 s, the same options occur:
Either from
f ” 0 pL1 q;

f ” 0 pN1 q, that is f ” a2 ¨ b2 pKq

it always follows that
f ” l1 ¨ b2 ;

l1 ” 0 pL1 q;

and so by (4’):
K “ K1 .
Or there is at least one f for which there exists no such l1 , which leads
2
to the construction of L2 “ pL1 , a2 q, N2 “ pK, b2 q, with L2 ¨ N2 ” 0 pKq,
where L2 is a proper divisor of L1 . Therefore, continuing in this way, in
general we define
L0 “ pK, aq; L1 “ pL0 , a1 q; . . . ; Lν “ pLν´1 , aν q; . . . ;
ν

N0 “ pK, bq; N1 “ pK, b2 q; . . . ; Nν “ pK, b2 q; . . . ,
where the ai are defined so that there exists an f such that
f ” 0 pLi´1 q;

f ” 0 pNi´1 q,

that is
i´1

f ” ai ¨ b2

pKq;

but ai ı 0 pLi´1 q.

Subsequently it holds in general that Li ¨ Ni ” 0 pKq, that by (3) Ni is a
proper divisor of K, and that Li is a proper divisor of Li´1 . By Theorem I
of the Finite Chain, the chain of the L must therefore terminate in finitely
many steps, say at Ln . For each f ” 0 pLn q, f ” 0 pNn q, it follows that
n
f ” ln ¨ b2 pKq, with ln ” 0 pLn q, and consequently by the above conclusion
K “ rLn , Nn s, which proves that K is reducible.
As a result of the preceding proofs, the uniqueness of the associated
prime ideals emerges as follows:
20

Let
M “ rB1 , . . . , Bk s “ rD1 , . . . , Dk s
be two shortest, and therefore reduced, representations of M as the least
common multiple of irreducible ideals, the numbers of components of which
are equal by Theorem IV. Then, by this theorem, the intermediate representations
M “ rD1 , . . . , Di´1 , Bi , Bi`1 , . . . , Bk s “ rD1 , . . . , Di´1 , Di , Bi`1 , . . . , Bk s
“ rĀi , Bi s “ rĀi , Di s
occurring there (where, as was remarked there, the index ji “ i can be set)
are also shortest representations. It therefore comes about that
Āi ¨ Bi ” 0 pDi q, Āi ı 0 pDi q;

Āi ¨ Di ” 0 pBi q, Āi ı 0 pBi q.

Because now by Theorem IV the irreducible ideals Bi and Di are primary,
it follows that there exist two numbers λi and µi such that
Dµi i ” 0 pBi q.

Bλi i ” 0 pDi q;

(5)

Now let Pi and P̄i denote the corresponding prime ideals of Bi and Di
respectively; therefore P̺i i ” 0 pBi q and P̄σi i ” 0 pDi q, and so by (5)
Piλi ̺i ” 0 pP̄i q;

P̄iµi σi ” 0 pPi q,

and from this, by the property of prime ideals:
Pi ” 0 pP̄i q;

P̄i ” 0 pPi q;

Pi “ P̄i .

With this we have proven
Theorem VII. For two different shortest representations of an ideal as
the least common multiple of irreducible ideals, the associated prime ideals
agree, and in fact the same associated prime ideals 20 also occur for every
decomposition thereof. The ideals themselves can be paired up in at least one
way such that a power of the ideal Bi is divisible by the associated Di and
vice versa. The numbers of ideals agree by Theorem IV.21
20

This is shown, for instance, in the example from footnote 19:
px2 , xy, y λ q “ rpx2 , yq, px, y λ qs,

where λ ě 2. Both corresponding prime ideals are px, yq here.
21
For the uniqueness of the “isolated” ideals found among the irreducible ideals, see §7.

21

§5. Representation of an ideal as the least common
multiple of maximal primary ideals. Uniqueness of
the associated prime ideals.
Definition IV. A shortest representation M “ rQ1 , . . . , Qα s is called
the least common multiple of maximal primary ideals if all Q are primary,
but the least common multiple of two Q is no longer primary.
That at least one such representation always exists follows from the representation of M as the least common multiple of irreducible ideals. This is
because these ideals are primary; either there now already exists a representation through maximal primary ideals, or else the least common multiple
of some two ideals is again primary. Because taking this least common multiple decreases the number of ideals by one, continuation of this procedure
leads to the desired representation in finitely many steps.
This representation is reduced by Lemma IV. Conversely, every reduced
representation arises through maximal primary ideals in this way, as the
decomposition of the Q into irreducible ideals shows.
In order to achieve an appropriate theorem of uniqueness here from Theorem VII, the connection with the associated prime ideals must be investigated, via
Theorem VIII. Should the primary ideals N1 , N2 , . . . , Nλ all have the
same associated prime ideal P, then their least common multiple Q “
rN1 , N2 , . . . , Nλ s is also primary and has P as its associated prime ideal.
If conversely Q “ rN1 , . . . , Nλ s is a reduced representation for the primary
ideal Q, then all Ni are primary and have as their associated prime ideal
the associated prime ideal P of Q.
To prove the first part of the statement, first note that P̺i ” 0 pNi q for
each i also implies Pτ ” 0 pQq, where τ denotes the largest of the indices ̺i .
Because P is furthermore also a divisor of Q, P is necessarily the associated
prime ideal, if Q is primary. It follows from
A ¨ B ” 0 pQq;

Bk ı 0 pQq (for every k)

22

that consequently
B ı 0 pPq; so Bk ı pNi q (for every k)
and as a result
A ” 0 pNi q; so A ” 0 pQq,
which proves that Q is primary, and that P is the associated prime ideal.
Conversely, first let
Q “ rN1 , . . . , Nλ s “ rNi , Li s
be a shortest representation of Q (where Q is primary) using primary ideals
Ni , and let Pi be the respective associated prime ideals. It follows from
Li ¨ Ni ” 0 pQq;

Li ı 0 pQq

(because of the shortest representation) that
Nσi i ” 0 pQq; or P̺i i σi ” 0 pQq.
Because the Pi are all divisors of Q, Pi is therefore equal to the associated
prime ideal P of Q for all i.
It remains to show that for every reduced representation Q “ rN1 , . . . , Nλ s,
the Ni are primary.22 For this, decompose the Ni into their irreducible ideals B; in the resulting shortest representation Q “ rB1 , . . . , Bµ s, each ideal
is then primary and has P as its associated prime ideal by the previous
parts of the proof. The same then holds for each Ni by the first part of the
theorem, proved above, which completes the proof of Theorem VIII.
Remark. From this it follows that a prime ideal is necessarily irreducible. This is because the reduced representation P “ rN, Ls gives
N ” 0 pPq by Theorem VIII, and so, because P ” 0 pNq, this also gives
P “ N and similarly P “ L. The irreducibility of P also follows directly,
The example Q “ rx2 , xy, y λ s “ rpx2 , xy, y 2 , yzq, px, y λ qs, where λ ě 2, which is
not reduced, shows that the reduced representation is crucial here. Here px2 , xy, y 2 , yzq “
rpx2 , yq, px, y 2 , zqs is not primary by the above proof; this is because the last representation
is a shortest one through primary ideals, but the associated prime ideals px, yq and px, y, zq
are different. (Q is primary by footnote 19.)
22

23

because P “ rN, Ls implies N ¨ L ” 0 pPq, N ı pPq, L ı 0 pPq, in contradiction with the defining property of a prime ideal.
Now let
M “ rQ1 , . . . , Qα s “ rQ̄1 , . . . , Q̄β s
be two reduced representations of M as the least common multiple of maximal primary ideals. By decomposing the Q into their irreducible ideals B
and the Q̄ respectively into the irreducible ideals D, two reduced representations of M as the least common multiple of irreducible ideals are produced,
in which by Theorem VII both the numbers of components and the associated prime ideals are the same. By Theorem VIII, all irreducible ideals
B for a fixed Qi have the same associated prime ideal Pi , while the Pk
associated with Qk is necessarily different from this, because otherwise by
Theorem VIII no representation using maximal primary ideals exists. The
number α of Q is therefore equal to the number of different associated prime
ideals P of the B; these different P construct the associated prime ideals of
the Q. The same applies to the Q̄ with respect to their decomposition into
the D. From Theorem VII it therefore follows that the number of Q and
Q̄ are the same, and that their corresponding prime ideals are the same.
Together these show that the summary given at the start of the section
about the irreducible ideals among maximal primary ideals holds true; that
is, these, and only these, all have the same associated prime ideal. Theorem
VII further shows the property of irreducibility of the maximal primary ideals: they admit no reduced representation as the least common multiple of
maximal primary ideals.
In summary the following has been proved:
Theorem IX. For two reduced representations of an ideal as the least
common multiple of maximal primary ideals, the numbers of components
and the associated prime ideals (which are all different from each other) are
the same. In other words, each Q can be uniquely associated with a Q̄ so
that a power of Q is divisible by Q̄, and vice versa.23 The Q and Q̄ have
23

One example of different representations is that given in footnote 12 for Theorem II:
px2 , xyq “ rpxq, px2 , µx`yqs for arbitrary µ. Because the associated prime ideals P1 “ pxq,
P2 “ px, yq are different, it consists of maximal primary ideals. For the uniqueness of the
“isolated” maximal primary ideals, see §7.

24

the property of irreducibility with respect to the decomposition into maximal
primary ideals.
Remark. Note that Theorem IX remains for the most part true if instead of reduced representations, only shortest representations are required.
If for instance M “ rQ1 , . . . , Q˚i , . . . , Qα s is then reduced with respect to Q˚i ,
and Q˚i is a proper divisor of Qi , then by Lemma IV Qi “ rQ˚i , pLi , Qi qs,
where Li is the complement of Qi ; this representation is reduced with respect
to Q˚i . By Theorem VIII, where in this application pLi , Qi q is replaced with
a proper divisor if necessary, Q˚i is therefore primary and has the same associated prime ideal Pi as Qi . Continuation of this procedure shows that every
such representation can be assigned a reduced representation through maximal primary ideals such that the number of components and the associated
prime ideals are the same. It therefore also holds for shortest representations that in two different representations the number of components and the
associated prime ideals are the same.
The thus uniquely defined associated prime ideals that are different from
each other shall be called “the associated prime ideals of M” for short.

§6. Unique representation of an ideal as the least
common multiple of relatively prime irreducible ideals.
Definition V. An ideal R is called relatively prime to S if T¨R ” 0 pSq
necessarily implies T ” 0 pSq. If R is relatively prime to S and S is also
relatively prime to R, then R and S are called mutually relatively prime.24
An ideal is called relatively prime irreducible if it cannot be expressed as the
least common multiple of mutually relatively prime proper divisors.
In particular, if instead of T the greatest common divisor T0 of all T
for which T ¨ R ” 0 pSq is taken, then we also have T0 ¨ R ” 0 pSq and
S ” 0 pT0 q. Therefore T0 “ S if R is relatively prime to S, and T0 is a
24

The relation of being relatively prime is not symmetric. For example, R “ px2 , yq is
relatively prime to S “ pxq, but S is not relatively prime to R, because S2 ” 0 pRq,
whereas T “ S ı 0 pRq.

25

proper divisor of S if R is not relatively prime to S.25
The proof of uniqueness is underpinned by
Theorem X. 1. If R is relatively prime to the ideals S1 , . . . , Sλ , then
R is also relatively prime to their least common multiple S.
2. If the ideals S1 , . . . , Sλ are relatively prime to R, then their least
common multiple S is also relatively prime to R.
3. If R is relatively prime to S and S “ rS1 , . . . , Sλ s is a reduced
representation of S, then R is also relatively prime to each Si .
4. If S is relatively prime to R, then each divisor Si of S is also relatively prime to R.
1. Because T ¨ R ” 0 pSq necessarily implies T ¨ R ” 0 pSi q, our
assumption implies T ” 0 pSi q, and therefore T ” 0 pSq.
2. Let C1 “ rS2 , . . . , Sλ s, C12 “ rS3 , . . . , Sλ s, . . . , C12...λ´1 “ Sλ . From
T¨S ” 0 pRq it then follows that T¨C1 ¨S1 ” 0 pRq, and so by our assumption
T ¨ C1 ” 0 pRq. From this it follows furthermore that T ¨ C12 ¨ S2 ” 0 pRq,
and so by our assumption T ¨ C12 ” 0 pRq. Proceeding in this way, it follows
finally that T ¨ C12...λ´1 “ T ¨ Sλ ” 0 pRq, and so T ” 0 pRq.
3. Let Ci denote the complement of Si ; it then follows from T0 ¨ R ”
0 pSi q that rT0 , Ci s ¨ R ” 0 pSq, since Ci ¨ R ” 0 pCi q.
Should T0 now be a proper divisor of Si , then because of the reduced
representation S “ rSi , Ci s, it is also true that rT0 , Ci s is a proper divisor
of S, contradicting our assumption.
4. It follows from T0 ¨ Si ” 0 pRq, where T0 is a proper divisor of R,
that T0 ¨ S ” 0 pRq; this contradicts our assumption.
Definition V shows in particular that every ideal is relatively prime to
the trivial ideal O consisting of all elements of Σ;26 the following theorems
however apply only for ideals other than O.
25

This so defined T0 is the same as Lasker’s “residual module” and Dedekind’s “quotient” of two modules in his expansion into modules rather than ideals. Lasker, loc. cit.
p49, Dedekind (Zahlentheorie), p504.
26
O plays the role of the trivial ideal only with respect to divisibility and least common
multiples, not with respect to the formation of products. For example, O “ pxq for the
ring of all polynomials in x with integer coefficients and without a constant term; O “ p2q
for the ring of all even numbers.

26

From Theorem X, which has just been proved, it immediately follows
that:
Lemma V. Every representation of an ideal as the least common multiple of mutually relatively prime ideals other than O is reduced.
Let M “ rR1 , R2 , . . . , Rσ s “ rRi , Li s be one such representation. By
Theorem X, part 2, Li is then also relatively prime to Ri ; Ri can therefore
not appear in Li , and so the representation is a shortest one. This is because
if Li ” 0 pRi q, where Li is relatively prime to Ri , it would also follow that
O ” 0 pRi q, as O ¨ Li ” 0 pRi q, and it therefore holds that Ri “ O, which is
by assumption impossible. Now suppose Ri can be replaced with the proper
divisor R˚i . Then Ri is reducible by Lemma II, and Ri “ rR˚i , pRi , Li qs. Replace pRi , Li q here with a proper divisor pRi , Li q˚ if necessary, and likewise
replace R˚i with a proper divisor if necessary so that a reduced representation for Ri is formed. Then Theorem X, part 3, shows that Li is also
relatively prime to pRi , Li q˚ , and this is different from O because of the
shortest representation. Because however pRi , Li q˚ appears in pRi , Li q and
pRi , Li q appears in Li , a contradiction therefore arises.
From Theorem X a further theorem arises, concerning the connection
with the associated prime ideals:
Theorem XI. If R is relatively prime to S and S is different from
O, then no associated prime ideal of R27 is divisible by an associated prime
ideal of S. Conversely, should no such divisibility occur, then R is relatively
prime to S, and of course S is different from O.
Let
R “ rQ1 , . . . , Qα s,

S “ rQ˚1 , . . . , Q˚β s

be reduced representations of R and S through maximal primary ideals,
and let P1 , . . . , Pα , P˚1 , . . . , P˚β be the associated prime ideals. We prove
the statement in the following form: if a P is divisible by a P˚ , then R
cannot be relatively prime to S, and vice versa.
27

The definition of associated prime ideals of R is given in the conclusion of §5.

27

Therefore let Pµ ” 0 pP˚ν q and as a result also Qµ ” 0 pP˚ν q, from
which by definition of P˚ν it follows that Qσµν ” 0 pQ˚ν q, and therefore also
Rσν ” 0 pQ˚ν q. Now let Rτ be the lowest power of R divisible by Q˚ν . If
τ “ 1, then R is divisible by Q˚ν ; because, however, S is different from O
by assumption, R is therefore not relatively prime to Q˚ν . If τ ě 2, then
Rτ ´1 ¨R ” 0 pQ˚ν q, Rτ ´1 ı 0 pQ˚ν q, and so R is not relatively prime to Q˚ν ,28
and hence in both cases R is also not relatively prime to S by Theorem X,
part 3.
If conversely R is not relatively prime to S, then by Theorem X, part
1, R is also not relatively prime to at least one Q˚ν . It therefore holds that
T0 ¨ R ” 0 pQ˚ν q, T0 ı 0 pQ˚ν q, and from this, because Q˚ν is primary, it
follows that Rτ ” 0 pQ˚ν q, and so also Qτ1 . . . Qτα ” 0 pQ˚ν q. It then follows,
however, that Pτ1 ̺1 . . . Pτα̺α ” 0 pP˚ν q for the associated prime ideals, and
by the properties of the prime ideals, P˚ν is therefore contained in at least
one P, which completes the proof of Theorem XI.
The existence29 and uniqueness of the decomposition into relatively prime
irreducible ideals now arises from Theorems X and XI as follows:
Let M “ rQ1 , . . . , Qα s be a reduced (or at least shortest) representation
of M through maximal primary ideals, and let P1 , . . . , Pα be the associated
prime ideals. We collect the P together in groups such that no ideal in a
group is divisible by an ideal in a different group, and each individual group
cannot be split into two subgroups that both have this property. In order to
construct such a grouping, it must be noted that by definition the group G
containing each ideal P must also contain all its divisors and multiples (that
is to say, divisible by P) occurring in P1 . . . Pα . For example, let Ppi1 q be all
pi q
pi ,i q
the multiples of P, Pj11 all the divisors of Ppi1 q , Pj11 2 all the multiples of
pi q

pi ...i q

pi ...i q

pi ...i q

pi ...i

q

λ´1
λ
and
is all the multiples of Pj11...jλ´1
Pj11 and so on; so in general Pj11...jλ´1
λ
. Because it deals with only finitely
Pj11...jλλ is all the divisors of Pj11...jλ´1
many ideals P, this algorithm must terminate in finitely many steps; that
is, no ideal different from all preceding ones is left over as a result. The thus
28

R0 is not defined, because Σ does not need to contain a unit; therefore the case τ “ 1
must be considered separately. τ “ 0 is also impossible when Σ does contain a unit, due
to the assumption regarding S.
29
The existence of the decomposition can also be proved directly, in exact analogy with
the proof of the existence of the decomposition into finitely many irreducible ideals given
in §2.

28

obtained system of ideals P now in fact forms a group G with the desired
properties.
pi ...iλ´1 q
all multiples
By definition G contains in addition to each Pj11...jλ´1
pi ...i q

pi ...i q

pi ...i q

λ
λ
all divisors Pj11...jλλ . Among
, and in addition to each Pj11...jλ´1
Pj11...jλ´1

pi ...i

q

λ´1
, while the
these however are also contained all divisors of the Pj11...jλ´1

pi ...i q

pi ...i q

λ
λ
. The ideals not
are themselves again Pj11...jλ´1
multiples of the Pj11...jλ´1
contained in G can therefore neither be divisors nor multiples of those ideals
contained in G. G however also satisfies the irreducibility condition. Because
if a division into two subgroups Gp1q and Gp2q exists, and Gp1q contains for
pi ...iλ q
pi ...i q
), it then also contains all preceding
instance Pj11...jλλ (respectively Pj11...jλ´1
ideals, because these are alternating multiples and divisors (respectively
divisors and multiples). So it also contains P, and thus the whole group
G. Should we continue accordingly with the ideals not contained in G,
then a division of all P into groups G1 , . . . , Gσ which possess the desired
1
1
properties is obtained. Such a grouping is unique; because if G1 , . . . , Gτ is
1
pi ...iλ q
pi ...i q
) is an element of Gi ,
a second grouping and Pj11...jλλ (respectively Pj11...jλ´1
1
then by the above reasoning Gi contains the whole group G and is therefore
identically equal to G because of the irreducibility property.
Now denote by Piµ (where µ runs from 1 to λi ) the ideals P collected
in a group Gi . Then, because the P are all different from each other, the
associated primary ideals Qiµ of a shortest representation are uniquely determined. Should we set

Ri “ rQi1 , . . . , Qiλi s, then M “ rR1 , . . . , Rσ s;
we show that, as a result, a decomposition of M into relatively prime irreducible ideals is achieved. First of all, it must be noted that Theorem XI
also remains applicable when Ri “ rQi1 , . . . , Qiλi s is only a shortest representation, because the associated prime ideals are uniquely defined, as seen
from the remark on Theorem IX. Because now no associated prime ideal Piµ
of Ri is divisible by an associated prime ideal Pjν of Rj and vice versa, by
Theorem XI Ri and Rj are mutually relatively prime, and each individual
Ri is relatively prime irreducible by Theorem XI due to the irreducibility
property of the group Gi , because with reducibility, ideals completely different from O come into question. Furthermore, by Lemma V it is a reduced
representation also if we had originally only started out with a shortest representation through the Q. Conversely, every decomposition into relatively
29

prime irreducible ideals leads to the given grouping of the Piµ by Theorem
XI.
Now let M “ rR̄1 , . . . , R̄τ s be a second representation of M through
relatively prime irreducible ideals, which is reduced by Lemma V. Because
then, as the decomposition of the R̄ into maximal primary ideals shows, the
associated prime ideals are the same, the groupings of these prime ideals,
the uniqueness of which was proven above, are therefore also the same. It
therefore follows that τ “ σ; and the notation can be chosen so that Ri and
R̄i belong to the same group. Therefore let
M “ rRi , Li s “ rR̄i , L̄i s
be the representation through ideal and complement. Then, because R̄i is
associated with the same group as Ri , by Theorem XI Li is also relatively
prime to R̄i and L̄i relatively prime to Ri . Because
Ri ¨ Li ” 0 pR̄i q,

R̄i ¨ L̄i ” 0 pRi q

it therefore follows that
Ri ” 0 pR̄i q;

R̄i ” 0 pRi q;

Ri “ R̄i .

Hence we have proved
Theorem XII. Every ideal can be uniquely expressed as the least common multiple of finitely many mutually relatively prime and relatively prime
irreducible ideals.

§7. Uniqueness of the isolated ideals.
Definition VI. If the shortest representation M “ rR, Ls is reduced
with respect to L, then R is called an isolated ideal if no associated prime
ideal of R appears in an associated prime ideal of L, or in other words, if L
is relatively prime to R.
Subsequently the representation M “ rR, Ls satisfies the requirements
of the representation M “ rRi , Li s in Lemma V, and with Lemma V we
have proved

30

Lemma VI. If R is an isolated ideal of the shortest and reduced (with
respect to L) representation M “ rR, Ls, then the representation is also reduced with respect to R.
Because we therefore always have a reduced representation when using
isolated ideals, the associated prime ideals occurring in the decompositions
of R and L into irreducible ideals complement each other to give the uniquely
determined associated prime ideals occurring in the corresponding decomposition of M.
Therefore no prime ideal of R belonging to the decomposition into irreducible ideals appears in the remaining prime ideals belonging to the corresponding decomposition of M. If conversely this condition is satisfied,
and if R features in at least one representation M “ rR, Ls reduced with
respect to R, and so also in a reduced representation M “ rR, L˚ s, then R
is isolated by Definition VI. From this the following definition arises, which
is independent of the particular complement L:
Definition VIa. R is called an isolated ideal if the prime ideals of R
belonging to the decomposition into irreducible ideals do not appear in the
remaining prime ideals belonging to the corresponding decomposition of M,
and if R appears in at least one representation M “ rR, Ls reduced with
respect to R.30
Now let
M “ rR, Ls “ rR̄, L̄s
be two representations of M through isolated ideals R and R̄ and complements L and L̄ such that the associated prime ideals of R and R̄ are the
same. Should L and L̄ be replaced with divisors L˚ and L̄˚ such that the
representations are reduced, then the associated prime ideals of L˚ and L̄˚
are also the same; by Theorem XI L˚ is therefore relatively prime to R̄ and
30

Should ordinary associated prime ideals (conclusion of §5) be introduced, then it
would be added as a special requirement that those of L all be different from those of R.
Following Definition VIa the representation therefore needs no longer be assumed to be
reduced with respect to the complement.

31

L̄˚ relatively prime to R. Because
R ¨ L˚ ” 0 pR̄q;

R̄ ¨ L̄˚ ” 0 pRq

it therefore follows that
R ” 0 pR̄q;

R̄ ” 0 pRq;

R “ R̄;

isolated ideals are therefore uniquely determined by the associated prime
ideals. This in particular results in a strengthening of Theorems VII and IX
regarding the decomposition into irreducible and maximal primary ideals,
where due to the remark on Theorem IX only shortest representations need
to be assumed. In summary:
Theorem XIII. For each shortest representation of an ideal as the least
common multiple of irreducible ideals (respectively maximal primary ideals), the isolated irreducible ideals (respectively maximal primary ideals) are
uniquely determined; the non-uniqueness applies only to the non-isolated
irreducible ideals (respectively maximal primary ideals).31 In general, the
isolated ideals are uniquely determined by the associated prime ideals.
If the ideals Bi (respectively Dj ) in one such shortest representation
through irreducible ideals (respectively maximal primary ideals) are nonisolated, then by definition the complements Ai (respectively Lj ) are divisible by Pi (respectively Pj ). It therefore follows that
σ

A̺i i ” 0 pBi q (respectively Lj j ” 0 pDj q).
It follows conversely from the fulfilment of these relations that Pi (respectively Pj ) appears in at least one associated prime ideal of the complement,
and so by the remark on Theorem IX also in an associated prime ideal of the
divisor L˚j of Lj that leads to a representation that is reduced with respect
to L˚j ; the Bi (respectively Dj ) are therefore non-isolated. In particular, irreducible ideals Bi for which the associated prime ideal appears more than
once in the decomposition of M are always non-isolated. Non-isolated primary ideals are therefore also characterised by the fact that a power of each
31

This theorem is already given without proof by Macaulay for polynomial ideals in the
case of the decomposition into maximal primary ideals; his definition of the isolated and
non-isolated (imbedded) primary ideals can be viewed as the irrational version of that
given below.

32

complement is divisible by them; isolated primary ideals are characterised by
the fact that this cannot be satisfied.

§8. Unique representation of an ideal as the product of coprime irreducible ideals.
Should the ring Σ contain a unit, that is, an element ε such that ε¨a “ a
for all elements in Σ,32 then the coprime ideals can be defined by
Definition VIII. Two ideals R and S are called coprime if their greatest common divisor is the trivial ideal O “ pεq consisting of all elements of
Σ. An ideal is called coprime irreducible if it cannot be expressed as the least
common multiple of pairwise coprime ideals.
Note that two coprime ideals are always mutually relatively prime. By
definition there are two elements r ” 0 pRq and s ” 0 pSq such that ε “
r ` s. It follows from T ¨ R ” 0 pSq however that T ¨ r ” 0 pSq, and so
T ¨ ε “ T ” 0 pSq; similarly T̄ ¨ S ” 0 pRq gives T̄ ” 0 pRq.33 It therefore
follows from Lemma V that each representation through pairwise coprime
ideals is reduced.
The following theorem, analogous to Theorem X, lays the foundation for
the proof of uniqueness:
Theorem XIV. If R is coprime to each of the ideals S1 , . . . , Sλ , then
R is also coprime to S “ rS1 , . . . , Sλ s. Conversely, it also follows from the
coprimality of R and S that R is coprime to each Sj . If R “ rR1 , . . . , Rµ s
and each Ri is coprime to each Sj , then R and S are coprime; the converse
again holds here.
If R is coprime to each Sj , then there exist elements sj such that
sj ” 0 pSj q;
32

sj ” ε pRq.

Σ clearly cannot contain more than one unit because of the commutativity of multiplication, since for some two units ε1 and ε2 it holds that ε1 ε2 “ ε2 “ ε1 .
33
The converse does not hold, however; for example, the ideals R “ pxq and S “ pyq
are mutually relatively prime, but not coprime.

33

Hence
s1 ¨ s2 ¨ . . . ¨ sλ ” 0 pSq;

s1 ¨ s2 ¨ . . . ¨ sλ ” ε pRq,

pR, Sq “ pεq.

Because, however, pR, Sq is divisible by each pR, Sj q, the converse also
holds. Repeated application of this conclusion results in the second part of
the statement. Namely, if Ri is coprime to S1 , . . . , Sλ for fixed i, then Ri
is coprime to S. If this holds for every i, then because the relationship of
coprimality is symmetric, S is coprime to R. Conversely, the coprimality of
S to Ri follows from the coprimality of R and S, and from this follows the
coprimality of Ri to Sj .
The proof of the existence and uniqueness34 of the decomposition into
coprime irreducible ideals comes from a unique grouping, like the corresponding proof for relatively prime ideals. Because, however, the relatively
prime irreducible ideals R1 , . . . , Rσ of M are uniquely defined by Theorem
XII, referring back to the associated prime ideals is unnecessary here.
We collect together the uniquely defined relatively prime irreducible ideals R1 , . . . , Rσ of M in groups in such a way that each ideal of a group
is coprime to every ideal of a group different from it, while each individual
group cannot be split into two subgroups such that each ideal of a subgroup is
coprime to every ideal of the other subgroup. One such grouping is given as
follows: by definition, for every ideal R contained in each individual group
G, all ideals not coprime to R must also be contained in G. For instance,
let these be denoted by Ri1 ; let Ri1 i2 be not coprime to these; in general, let
Ri1 ...iλ be not coprime to Ri1 ...iλ´1 . Because we are only dealing with finitely
many ideals in total, this procedure must terminate in finitely many steps,
that is, there comes a point after which no ideals different from all those preceding are yielded; the thus obtained system of ideals contructs a group G
with the desired properties, because all ideals not contained in G are by construction coprime to all those contained in G. Suppose there further exists
a splitting into two subgroups Gp1q and Gp2q ; and without loss of generality
let Ri1 ...iλ be an element of Gp1q . Because the relationship of coprimality
is symmetric, Gp1q must then also contain Ri1 ...iλ´1 , . . . , Ri1 , and so also R
and consequently the whole group G, which proves irreducibility. If we then
34

The existence of the decomposition can again be proved in direct analogy with §2; the
proof of uniqueness can also be conducted directly (cf. that mentioned in the introduction
about Schmeidler and Noether-Schmeidler). The proof given here also gives an insight
into the structure of the coprime irreducible ideals.

34

proceed accordingly with the groups not contained in G, then we obtain a
1
1
grouping G1 , . . . , Gτ of all R. This grouping is unique; let G1 , . . . , Gτ be a
1
1
second grouping and Ri1 ...iλ an element of Gi . Then Gi also contains R and
hence G1 , and because of the irreducibility requirement cannot contain any
1
elements different from G1 ; thus Gi is the same as G1 .
Now let Ti be the least common multiple of the ideals R incorporated
into a group Gi . We show that M “ rT1 , . . . , Tτ s is a representation of
M through coprime irreducible ideals. First of all, the decomposition of the
T into the R shows that rT1 , . . . , Tτ s really does give a represention of M.
Furthermore, by Theorem XIV the T are pairwise coprime, and each T is
coprime irreducible. Hence the existence of such a representation is proven.
For the proof of uniqueness, let M “ rT̄1 , . . . , T̄τ̄ s be a second such
representation. If we decompose the T̄ into their relatively prime irreducible
ideals R, then the R appearing in different T̄ are coprime to each other by
Theorem XIV, and so are also mutually relatively prime; they are therefore
the same as the uniquely defined relatively prime irreducible ideals R of M.
1
By Theorem XIV, the T̄i further generate a grouping Gi of the R with the
given properties. Because, however, this grouping is unique, and every T̄i
1
is uniquely defined by the group Gi “ Gi , it follows that T̄i “ Ti , proving
uniqueness.
Furthermore, for pairwise coprime ideals the least common multiple is
the same as the product.
Because by Theorem XIV the complement Li is also coprime to Ti for
M “ rT1 . . . Tτ s, there therefore exist elements
ti ” 0 pTi q;

li ” 0 pLi q;

ε “ ti ` li .

It therefore follows from f ” 0 pTi q, f ” 0 pLi q that, because
f “ f ε “ f ti ` f li , it also holds that f ” 0 pLi ¨ Ti q.
Because conversely Li ¨ Ti is divisible by rLi , Ti s, it follows that rLi , Ti s “
Li ¨ Ti , and by extension of the procedure over the Li , we conclude that
M “ rT1 , . . . , Tτ s “ T1 ¨ T2 ¨ . . . ¨ Tτ .
We have therefore proved
Theorem XV. Every ideal can be uniquely expressed as the product of
finitely many pairwise coprime and coprime irreducible ideals.
35

§9. Development of the study of modules. Equality of the number of components in decompositions
into irreducible modules.
We now show that the content of the first three sections, which relates
to irreducible ideals, not primary and prime ideals, still holds under less
restrictive conditions. These sections in particular do not use the law of
commutativity of multiplication and concern only the property of ideals
being modules, and so remain upheld for modules over non-commutative
rings, which are now to be defined. The definition of these modules shall be
based upon a double domain pΣ, T q with the following properties:
Σ is an abstractly defined non-commutative ring, that is, Σ is a system
of elements a, b, c, . . . , for which two operations are defined; ring addition
Ś
(#) and ring multiplication ( ), which satisfy the laws set out in §1, with
the exception of law 4. regarding the commutativity of ring multiplication.
T is a system of elements α, β, γ, . . . for which, in conjunction with Σ,
two operations are also defined; addition, which for every two elements α, β
uniquely generates a third α ` β, and multiplication of an element α of T
with an element c of Σ, which uniquely generates an element c ¨ α of T .35
The following laws apply to these operations:
1. The law of associativity of addition: pα ` βq ` γ “ α ` pβ ` γq.
2. The law of commutativity of addition: α ` β “ β ` α.
3. The law of unrestricted and unique subtraction: there exists one and
only one element ξ in T which satisfies the equation α ` ξ “ β (written
ξ “ β ´ α).
Ś
4. The law of associativity of multiplication: a ¨ pb ¨ γq “ pa bq ¨ γ.
5. The law of distributivity: pa#bq ¨ γ “ a ¨ γ ` b ¨ γ; c ¨ pα ` βq “ c ¨ α ` c ¨ β.
The existence of the zero element follows from these conditions, as is
well-known, and also the validity of the law of distributivity for subtraction
and multiplication:
pa ´ bq ¨ γ “ a ¨ γ ´ b ¨ γ;

c ¨ pα ´ βq “ c ¨ α ´ c ¨ β;

35

We are therefore dealing with “right” multiplication, a “right” domain T , and thus
“right” modules and ideals. Were we to have based T on a left multiplication α ¨ c, then
a corresponding theory of the left modules and ideals would follow; M would contain in
Ś
addition to α also α ¨ c. The law of associativity would have the form pγ ¨ bq ¨ a “ γ ¨ pb aq
here.

36

where (´) denotes subtraction in Σ. If Σ contains a unit ε, then ε ¨ α “ α
holds for all elements α of T .
Let a module M over pΣ, T q be understood to be a system of elements
of T which satisfies the following two conditions:
1. For each element α of M , c ¨ α is also an element of M , where c is an
arbitrary element of Σ.
2. For each pair of elements α and β of M , the difference α ´ β is also an
element of M ; therefore for each element α of M , nα is also an element of
M for every integer n.36
By this definition, T itself constitutes a module in pΣ, T q. If in particular
T and the operations defined for it coincide with the ring Σ and the operations applicable to it, then the module M becomes a (right) ideal M in Σ.
If Σ is taken to be commutative, then the usual concept of an ideal arises,
which therefore comes about as a special case of the concept of a module.37
All definitions in §1 remain upheld for modules: so α ” 0 pM q and
N ” 0 pM q mean that α and each element of N respectively are in M ;
in other words, α and N respectively are divisible by M . M is a proper
divisor of N if M contains elements not in N ; it follows from N ” 0 pM q,
M ” 0 pN q that M “ N . The definitions of the greatest common divisor and
the least common multiple remain upheld word for word. In particular, if
M contains a finite number of elements α1 . . . α̺ such that M “ pα1 . . . α̺q,
that is α “ c1 α1 ` ¨ ¨ ¨ ` c̺ α̺ ` n1 α1 ` . . . n̺ α̺ for every α ” 0 pM q, where
the ci are elements of Σ and the ni are integers, then M is called a finite
module, and α1 . . . α̺ a module basis.
In the following we now use, analogously to §1, only domains pΣ, T q that
satisfy the finiteness condition: every module in pΣ, T q is finite, and so has
a module basis.
36

These integers are again to be considered as abbreviatory symbols, not as ring elements.
37
The simplest example of a module is the module consisting of integer linear forms;
here Σ consists of all integers and T consists of all integer linear forms. A somewhat
more general module arises if algebraic integers are used instead of integers in Σ and T ,
or for instance all even numbers. If we consider the complex of all coefficients as one
element each time instead of the linear forms, then the operations in Σ and T are in
fact different. Ideals in non-commutative rings of polynomials form the subject matter
of the joint work of Noether-Schmeidler. The lectures of Hurwitz on the number theory
of quaternions (Berlin, Springer 1919) and the works of Du Pasquier cited there relate to
ideals in further special non-commutative rings.

37

Theorem I of the Finite Chain then also holds for modules for this domain pΣ, T q, as the proof there shows, and thus all requirements for §§2 and
3 are satisfied. It remains to directly carry over Definition I and Lemma I
regarding shortest and reduced representations, and likewise Theorem II regarding the ability to represent each module as the least common multiple of
finitely many irreducible modules, whereby Lemma II shows that each such
shortest representation is immediately reduced. Furthermore, Theorem III
also remains upheld, which conveys the reducibility of a module through the
properties of its complement; and from that follows Lemma III, and finally
Theorem IV, which states the equality of the number of components in two
different shortest representations of a module as the least common multiple
of irreducible modules. Theorem IV then yields Lemma IV as the converse
of Lemma I on reduced representations.
Remark. The same reasoning shows that all of these theorems and definitions remain upheld if we understand all modules to be two-sided (that
is, if α is an element, then c ¨ α and α ¨ c are also elements, and if α and β
are elements, then α ´ β is also an element) for two-sided domains T , that
is, domains that are both left domains and right domains.
While all these theorems are based only on the notions of divisibility and
the least common multiple, the further theorems of uniqueness are based crucially on the concept of product, and therefore do not have a direct translation. Therefore the definitions of primary and prime ideals do not translate
to modules, because the product of two elements of T is not defined. Although it is feasible to formally construct a translation to non-commutative
rings, it loses its meaning, because here the existence of the associated prime
ideals cannot be proved,38 and the proof that an irreducible ideal is primary
also breaks down. However, these two cases lay the foundations for the
38

For example, from
pλ1 1 ” 0 pQq,

pλ2 2 ” 0 pQq

it does not follow that

pp1 ´ p2 qλ1 `λ2 ” 0 pQq

and likewise from
pa ¨ bqλ ” 0 pQq

it does not follow that

aλ ¨ bλ ” 0 pQq.

As a result, P cannot be proven to be an ideal, nor does it have the properties of prime
ideals. For two-sided ideals, P can be proven to be an ideal, but still not a prime ideal.

38

uniqueness theorems – in contrast, if the non-commutative ring contains a
unit, coprime and coprime irreducible ideals can be defined, and the reasoning used in the proof of Theorem II shows that every ideal can be represented
as the least common multiple of finitely many pairwise coprime and coprime
irreducible ideals.39
Finally, we mention another criterion sufficient for the finiteness condition to be satisfied in pΣ, T q: if Σ contains a unit and satisfies the finiteness
condition, and if T is itself a finite module in pΣ, T q, then every module in
pΣ, T q is finite.
It follows from the existence of the unit in Σ namely that for
M “ pf1 , . . . , f̺ q,

f “ b̄1 f1 ` ¨ ¨ ¨ ` b̺̄ f̺ ` n1 f1 ` ¨ ¨ ¨ ` n̺ f̺ ,

where the b̄i are elements of Σ and the ni are integers, there is also a representation of the form f “ b1 f1 ` ¨ ¨ ¨ ` b̺ f̺ , where the bi are elements of Σ.
This is because fi “ εfi implies ni fi “ ni εfi , and since ni ε “ pε ` ¨ ¨ ¨ ` εq
belongs to Σ, it then follows that bi “ b̄i ` ni ε is an element of Σ. The
requirement regarding T now states that every element α of T has a representation
α “ ā1 τ1 ` ¨ ¨ ¨ ` āk τk ` n1 τ1 ` ¨ ¨ ¨ ` nk τk ,
and hence

α “ a1 τ1 ` ¨ ¨ ¨ ` ak τk ,

where the second representation arises from the first because ετi “ τi , as
above.
If the α now run through the elements of a module M in pΣ, T q, then the
coefficients ak of τk run through the elements of an ideal Mk in Σ; by the
p1q
p̺q
above, for each ak ” 0 pMk q, we also have ak “ b1 ak ` ¨ ¨ ¨ ` b̺ ak . If we let
piq
αpiq denote an element of M for which the coefficient of τk is equal to ak ,
then α ´ b1 αp1q ´ . . . ´ b̺ αp̺q is an element belonging to M which depends
only on τ1 , . . . , τk´1 . This procedure can be repeated on the collection of
1
these elements no longer containing τk , which construct a module M , so
that after finitely many repetitions the proposition is proved.
39

For special “completely reduced” ideals, uniqueness theorems can also be established
here; cf. the joint work of Noether-Schmeidler.

39

§10. Special case of the polynomial ring.
1. The ring Σ that we take as our starting point consists of all polynomials in x1 , . . . , xn with arbitrary complex coefficients, for which the finiteness
condition is satisfied due to Hilbert’s Module Basis Theorem (Math. Ann.,
vol. 36). This section concerns the connection of our theorems with the
known theorems of elimination theory and module theory.
This connection is established through the following special case of a
famous theorem of Hilbert’s:40
If f vanishes for every (finite) system of values of x1 , . . . , xn that is a
root of all polynomials of a prime ideal P (we call such a system a root of
P), then f is divisible by P. In other words, a prime ideal P consists of all
polynomials that vanish at these roots.41
If a product f ¨ g vanishes for all roots of P, then at least one factor vanishes; the roots form an irreducible algebraic figure. Conversely, should we
begin with this definition of the irreducible figure, then it follows that the collection of polynomials vanishing on an irreducible figure forms a prime ideal;
prime ideals and irreducible figures therefore correspond with each other bijectively. Furthermore, because Q ” 0 pPq and P̺ ” 0 pQq, the roots of
a primary ideal are the same as those of its associated prime ideal.42 The
representation of an ideal as the least common multiple of maximal primary
ideals therefore yields a partition of all roots of the ideal into irreducible
figures; and as Lasker has shown, the converse also holds. The proof of the
40

Über die vollen Invariantensysteme. Math. Ann. 42 (1893), §3, p313.
This special case can also be proven directly in the case of homogeneous forms, as
Lasker [Math. Ann. 60 (1905), p607] has shown, and then conversely this again implies
the Hilbert Theorem (in the homogeneous and the inhomogeneous case), which we can
state as follows: if an ideal R vanishes at all roots of M, then a power of R is divisible by
M. This theorem, and likewise the special case, only holds however if the codomain of the
x is algebraically closed, and therefore cannot follow from our theorems alone, but must
use the existence of roots; for instance, in the case that an ideal for which the only root is
x1 “ 0, . . . , xn “ 0 contains all products of powers of the x of a particular dimension. The
remaining proof can be simplified somewhat by using our theorems via Lasker. Lasker
must in particular also make use of the Hilbert Theorem for the proof of the decomposition
of an ideal into maximal primary ideals.
42
Macaulay (cf. Introduction) uses this property of a primary ideal having an irreducible
figure in the definition, while Lasker only incorporates the concept of the manifold of a
figure in the definition, which is otherwise abstractly defined. The primary ideals which
vanish only for x1 “ 0, . . . , xn “ 0 take a special place in Lasker.
41

40

uniqueness of the associated prime ideals therefore corresponds here with the
Fundamental Theorem of Elimination Theory regarding the unique decomposability of an algebraic figure into irreducible figures; it can serve as the
equivalent of this theorem of elimination theory for special polynomial rings
where no unique representation of the polynomial as the product of irreducible polynomials of the ring exists, and consequently also no elimination
theory.
The irreducible figures corresponding to the isolated primary ideals are
exactly those occurring in the “minimal resolvent”;43 since the roots of each
non-isolated maximal primary ideal are at the same time roots of at least one
isolated one, namely one for which the associated prime ideal is divisible by
that of the non-isolated ideal. The uniqueness of the isolated primary ideals
therefore yields new invariant multiplicities in the exponents. Additionally,
the uniqueness theorems for the decomposition of the primary ideals into
irreducible ideals can be viewed as an addendum to elimination theory, in
the sense of multiplicity.
Following these remarks, the different decompositions can be interpreted
in their relation to the algebraic figures. The pairwise coprime ideals correspond to figures that have no roots in common; for the mutually relatively
prime ideals, likewise no irreducible figure of one ideal has roots in common
with that of another; the maximal primary ideals vanish only in irreducible
figures which are all different from each other; for the decomposition into
irreducible ideals, the same irreducible figures can also appear repeatedly.
It should be noted that the ring of all homogeneous forms can also be
used in place of the general polynomial ring, since it is easy to convince
ourselves that the general theorems also remain upheld for the operations
valid there44 – the addition is only defined for forms of the same dimension.
A simple example of the four different decompositions – for which the
formulae below follow – is given, following the above remarks, by one straight
line and two further straight lines skew to it and intersecting each other, one
43

Cf. for example J. König, Einleitung in die allgemeine Theorie der algebraischen
Größen (Leipzig, Teubner, 1903), p235.
44
The following example shows that here inhomogeneous decompositions can also exist
as well as homogeneous ones in the case of non-uniqueness, however:
px3 , xy, y 3 q “ rpx3 , yq; py 3 , xqs “ rpxy, x3 , y 3 , x ` y 2 q; pxy, x3 , y 3 , y ` x2 qs.

41

of which contains a point of higher multiplicity other than the point of intersection. The decomposition into coprime-irreducible ideals corresponds to
the decomposition into the straight line and the figure skew to it; this figure
splits into the two straight lines it is composed of for the decomposition into
relatively prime irreducible ideals; the decomposition into maximal primary
ideals corresponds to a detachment of the point of higher multiplicity, while
the decomposition into irreducible ideals requires the removal of this point.
Should we take this point as the starting point, and the straight line
passing through it as the y-axis, the straight line intersecting this parallel
to the x-axis, and the straight line skew to it parallel to the z-axis, then one
such configuration is represented by the following irreducible ideals:45
B1 “ px ´ 1, yq;

B2 “ py ´ 1, zq;

B3 “ px, zq;

B4 “ px3 , y, zq;

B5 “ px2 , y 2 , zq.
The associated prime ideals are:
P1 “ B1 ;

P2 “ B2 ;

P3 “ B3 ;

P4 “ P5 “ px, y, zq.

The maximal primary ideals are:
Q1 “ B1 ;

Q2 “ B2 ;

Q3 “ B3 ;

Q4 “ rB4 , B5 s “ px3 , y 2 , x2 y, zq.

The relatively prime irreducible ideals are:
R1 “ Q1 ;

R2 “ Q2 ;

R3 “ rQ3 , Q4 s “ px3 , x2 y, xy 2 , zq.

The coprime irreducible ideals are:
S1 “ R1 ;

S2 “ rR2 , R3 s “ ppy ´ 1qx3 , py ´ 1qx2 y, py ´ 1qxy 2 , zq.

This produces the total ideal:
M “ rS1 , S2 s
“ S1 ¨ S2
“ ppx ´ 1qpy ´ 1qx3 , py ´ 1qx2 y, py ´ 1qxy 2 , px ´ 1qz, ypy ´ 1qx3 , yzq,
45

The first three of these are irreducible since they are prime ideals; B4 is irreducible
because it only has the divisors px2 , y, zq and px, y, zq; B5 is irreducible because every
divisor contains the polynomial xy.

42

which gives
1 “ ´py ´ 1qx3 ` py ´ 1qpx3 ´ 1q ` y,

py ´ 1qpx3 ´ 1q ` y ” 0 pS1 q;

´py ´ 1qx3 ” 0 pS2 q.
Here the ideals B1 , B2 , B3 are isolated, and so also uniquely determined in the decompositions into irreducible and maximal primary ideals. The ideals B4 and B5 , and respectively Q4 , are non-isolated; they
are not uniquely determined, but rather can be replaced for example with
D4 “ px3 , y ` λx2 , zq, D5 “ px2 ` µxy, y 2 , zq; similarly, Q4 can be replaced
with
Q̄4 “ px3 , x2 y, y 2 ` λxy, zq.
2. Similarly to the general (and the integer) polynomial ring, every
finite integral domain of polynomials also satisfies the finiteness condition46
– as Hilbert’s Module Basis Theorem shows – where the coefficients can be
assigned an arbitrary field. We shall give another example for the ring of
all even polynomials, as the simplest ring where, because x2 ¨ y 2 “ pxyq2 , no
unique factorisation of the polynomial into irreducible polynomials of the
ring exists. It uses the same configuration as in the above example, which
is now given through the irreducible ideals
B1 “ px2 ´ 1, xy, y 2 , yzq;

B2 “ py 2 ´ 1, xz, yz, z 2 q;

B3 “ px2 , xy, xz, yz, z 2 q;

B4 “ px4 , xy, y 2 , xz, yz, z 2 q;

B5 “ px2 , y 2 , xz, yz, z 2 q.
The associated prime ideals are:
P1 “ B1 ;

P2 “ B2 ;

P3 “ B3 ;

P4 “ P5 “ px2 , xy, y 2 , xz, yz, z 2 q.

It follows that P1 is a prime ideal because every polynomial of the ring
has the following form:
f ” φpz 2 q ` xzψpz 2 q pP1 q.
46

Conversely, if the finiteness condition is satisfied for a polynomial ring, and if each
polynomial has at least one representation where the factors of lower degree are in x, then
the ring is a finite integral domain.

43

Therefore let
f1 ” φ1 pz 2 q ` xzψ1 pz 2 q;

f2 ” φ2 pz 2 q ` xzψ2 pz 2 q,

and thus, since f1 ¨ f2 ” 0 pP1 q, we have the existence of the following
equations:
φ1 φ2 ` z 2 ψ1 ψ2 “ 0;

φ2 ψ1 ` φ1 ψ2 “ 0,

and so f1 ” 0 pP1 q or f2 ” 0 pP1 q.
We can show that P2 is a prime ideal in precisely the same way; P3 is
a prime ideal because every polynomial of the ring mod P3 is congruent to
a polynomial in y 2 ; P4 consists of all polynomials of the ring. It therefore
also follows that B1 , B2 and B3 are irreducible, as they are prime ideals;
B4 and B5 each have only the sole proper divisor P4 , and so are necessarily
irreducible.
From the irreducible ideals arise the maximal primary ideals:
Q1 “ B1 ; Q2 “ B2 ; Q3 “ B3 ; Q4 “ rB4 , B5 s “ px4 , x3 y, y 2 , xz, yz, z 2 q;
the relatively prime irreducible ideals:
R1 “ Q1 ;

R2 “ Q2 ;

R3 “ rQ3 , Q4 s “ px4 , x3 y, x2 y 2 , xy 3 , xz, yz, z 2 q,

the coprime irreducible ideals:
S1 “ R1
S2 “ rR2 , R3 s
“ ppy 2 ´ 1qx4 , py 2 ´ 1qx3 y, py 2 ´ 1qx2 y 2 , py 2 ´ 1qxy 3 , xz, yz, z 2 q.
As in the first example, it follows that
rB4 , B5 s “ rD4 , D5 s;
where D4 “ px4 , xy ` λx2 , . . . q and D5 “ px2 ` µxy, . . . q for λ ¨ µ ‰ 1; and
rQ3 , Q4 s “ rQ3 , Q̄4 s;

where Q̄4 “ px4 , x3 y, y 2 ` λxy, . . . q.

The remaining irreducible ideals (and maximal primary ideals respectively) B1 , B2 , B3 are uniquely determined as isolated ideals.

44

§11. Examples from number theory and the theory
of differential expressions.
1. Let the ring Σ consist of all even integers. Σ can then be bijectively
assigned to all of the integers, since each number 2a in Σ can be allocated
the number a. From this it immediately follows that every ideal in Σ is a
principal ideal p2aq, where in the basis representation 2c “ n ¨ 2a of each
element 2c of the ideal, the odd numbers n only amount to abbreviations
for finite sums.
The prime ideals of the ring are given by P0 “ O “ p2q and P “ p2pq,
where p is an odd prime number; therefore every prime ideal is divisible by
P0 , but by no other prime ideal. The primary ideals are given by Q̺0 “
p2 ¨ 2̺0 q and Q̺ “ p2p̺ q; they are at the same time irreducible ideals, and by
what has been said about prime ideals, each two corresponding to different
odd prime numbers are mutually relatively prime, but no Q is relatively
prime to any Q̺0 ,47 and so the Q̺0 are the only non-isolated primary ideals.
The unique decomposition of a into prime powers corresponds to the unique
representation of the ideal p2aq through maximal primary (and at the same
time irreducible) ideals:
p2aq “ rp2 ¨ 2̺0 q, p2p1 q̺1 , . . . , p2pα q̺α s;
therefore, contrary to the examples from the polynomial ring, the nonisolated maximal primary ideals are also uniquely determined. As for ̺0 “ 0,
it behaves also as a representation through mutually relatively prime ideals,
while for ̺0 ą 0 the ideal is relatively prime irreducible.
While the four different decompositions therefore coincide in the ring
of all integers, this is only the case here for the two decompositions into
maximal primary ideals and irreducible ideals on the one hand, and for the
decompositions into coprime irreducible ideals (every ideal is coprime irreducible, because the ring has no units) and relatively prime irreducible ideals
for ̺0 ą 0 on the other hand, while for ̺0 “ 0 the coprime irreducible and
relatively prime irreducible decompositions are different from each other.
At the same time, an example presents itself here. A prime ideal can be
divisible by another without having to be identical to it; more generally, the
In fact, it always follows from 2b ¨ 2p̺11 ” 0 p2p̺22 q for odd p1 ‰ p2 that 2b ” 0 p2p̺22 q;
however, 2b ¨ 2p̺ ” 0 p2 ¨ 2̺0 q only implies that 2b ” 2 ¨ 2̺0 ´1 p2 ¨ 2̺0 q.
47

45

factorisation does not follow from divisibility. The last one – as a consequence of the fact that the ring contains no units – is also the reason that
no unique factorisation of the numbers of the ring into irreducible numbers
of the ring exists, although each ideal is a principal ideal; the introduction of
the least common multiple therefore proves to be necessary here. It should
also be noted that the relationships remain exactly the same if instead of
all even numbers, we use all numbers divisible by a fixed prime number or
prime power.
However, irreducible and primary ideals are different as well if Σ consists
of all numbers divisible by a composite number g “ p1σ1 . . . pσν ν . Every ideal
is again a principal ideal pg ¨ aq, and the prime ideals are again given by
P0 “ O “ pgq, P “ pg ¨ pq, where p is a prime number different from the
prime numbers p appearing in g. In contrast, the irreducible ideals are given
by Bλi “ pg¨pλi i q, B̺ “ pg¨p̺ q; the primary ideals are given by Q̺ “ B̺ and
by the Qλ1 ...λν “ pg ¨ pλ1 1 . . . pλν ν q different from the irreducible ideals, where
the Bλi and Qλ1 ...λν all have the same associated prime ideal P0 “ pgq.
Uniqueness of the decomposition into irreducible ideals also holds here, and
consequently uniqueness of the decomposition into maximal primary ideals
too, and so the non-isolated ideals are again uniquely determined here too.
2. One example of a non-commutative ring is presented by the ideal theory in non-commutative polynomial rings discussed in the Noether-Schmeidler
paper. It concerns in particular “completely reducible” ideals, that is, ideals for which the components of the decomposition are pairwise coprime and
have no proper divisors; the components are therefore a fortiori irreducible.
The equality of the number of components in two different decompositions
therefore follows from §9, in addition to the isomorphism proved there. Thus
a consequence of the decomposition of systems of partial or ordinary linear
differential expressions arising as a special case of the paper is obtained,
which even appears not to have been remarked upon in the well-known case
of an ordinary linear differential expression.
Meanwhile, the system T of all cosets of a fixed ideal M together with
the non-commutative polynomial ring Σ gives a double domain pΣ, T q, where
T has the module property with respect to Σ, since the difference of two
cosets is again a coset, and likewise the product of a coset with an arbitrary
polynomial, whereas the product of two cosets does not exist (loc. cit.
§3). The systems of cosets denoted there as “subgroups” form examples of

46

modules in double domains pΣ, T q where the ring Σ is non-commutative.

§12. Example from elementary divisor theory.
This section deals with a concept of elementary divisor theory contingent on the general developments thereof, which is however itself presumed
to be known.
Let Σ be the ring of all integer matrices with n2 elements, for which addition and multiplication are defined in the conventional sense for matrices.
Σ is then a non-commutative ring; the ideals are thus in general one-sided,
and two-sided ideals only arise as a special case.48
We first show that every ideal is a principal ideal. In order to do this,
we allocate (for right ideals) each matrix A “ paik q a module
A “ pa11 ξ1 ` ¨ ¨ ¨ ` a1n ξn , . . . , an1 ξ1 ` ¨ ¨ ¨ ` ann ξn q
consisting of integer linear forms. Conversely, each matrix which gives a
basis of A corresponds to this module, so in addition to A, U A also corresponds to A, where U is unimodular. More generally, a module B which is
a multiple of A corresponds to the product P A. A single linear form from
A is given by a P which contains only one non-zero row.
Now let A1 , A2 , . . . , Aν , . . . be all the elements of an ideal M, with A1 ,
A2 , . . . , Aν , . . . the modules assigned to them, A the greatest common
divisor of these, and U A the most general matrix assigned to this module A.
To every individual linear form from A then corresponds a matrix P1 Ai1 `
¨ ¨ ¨ ` Pσ Aiσ by definition of the greatest common divisor, where as above
the P contain only one non-zero row. From this it follows that the matrix
A corresponding to a basis of A can also have such a representation, now
with general P , and thus is an element of M. Because furthermore every
module Ai is divisible by A, every matrix Ai is therefore divisible by A; A,
and in general U A, forms a basis of M. If we were to deal with left ideals,
then we would have to consider the columns of each matrix as the basis of a
module accordingly; each ideal is a principal ideal for which A being a basis
48

The ideal theory of these rings forms the subject of the papers of Du Pasquier: Zahlentheorie der Tettarionen, Dissertation Zürich, Vierteljahrsschr. d. Naturf. Ges. Zürich,
51 (1906). Zur Theorie der Tettarionenideale, ibid., 52 (1907). The content of the two
papers is the proof that each ideal is a principal ideal.

47

implies that AV is also a basis, where V is understood to be an arbitrary
unimodular matrix.
In the following we now use two-sided ideals in the context of elementary divisor theory, which in particular implies that if A belongs to the
ideal, then so does P AQ. Thus, by that shown above, the most general
basis of one such ideal is given by U AV , where U and V are unimodular.
The basis elements therefore run through a class of equivalent matrices,49
and a one-to-one relationship exists between the ideal and the class. Consequently, such a relationship also exists between the ideal and the elementary
divisor system pa1 |a2 | . . . |an q of the class, where the ai are known to be nonnegative integers, each of which divides the following one. The matrix of
the class occurring in the normal form induced by the elementary divisors
can therefore be understood as a special basis of the ideal; the divisibility
of the elementary divisors follows from that of the ideals, respectively the
classes, and vice versa.
Now, however, Du Pasquier has shown loc. cit. §11 that for each twosided ideal the rank is n and all elementary divisors are the same. In order
to be able to consider the case of general elementary divisors, we must
therefore start not from the ideals, but directly from the two-sided classes
(which shall be denoted by capital letters such as A, B, C, . . . ). A class
A “ U AV is therefore divisible by another class B here if A “ P BQ.
In general, the following statement holds: the least common multiple
(greatest common divisor) of two classes is obtained through the construction
of the least common multiple (greatest common divisor) of the corresponding
systems of elementary divisors.50
Let pa1 |a2 | . . . |an q, pb1 |b2 | . . . |bn q, pc1 |c2 | . . . |cn q be the systems of elementary divisors of A, B, C˚ respectively, where ci “ rai , bi s, and let
C “ rA, Bs. Then C˚ is divisible by A and B, and therefore by C, and
conversely the elementary divisors of C are divisible by those of C˚ , which
implies that C is divisible by C˚ , and thus C “ C˚ . The proof for the greatest
49

The right and left classes respectively corresponding to the basis elements of the onesided ideals.
50
In the paper Zur Theorie der Moduln, Math. Ann. 52 (1899), p1, E. Steinitz defines
the least common multiple (greatest common divisor) of classes through least common multiples and greatest common divisors of the systems of elementary divisors. Independently
of this, the least common multiple of classes is found as the “congruence composition” in
H. Brandt, Komposition der binären quadratischen Formen relativ einer Grundform, J. f.
M. 150 (1919), p1.

48

common divisors proceeds likewise.
The unique representation of the elementary divisors ai as the least common multiple of prime powers therefore corresponds to a representation of A
as the least common multiple of classes Q, the elementary divisors of which
are given by the powers of one prime number, or in symbols:
Q „ ppr1 |pr2 | . . . |pr̺ |0| . . . |0q;

r1 ď r2 ď ¨ ¨ ¨ ď r̺ .

If in particular the rank ̺ is equal to n, then we have a decomposition into
coprime and coprime irreducible classes here, which, although the ring is
non-commutative, is unique.
The classes Q can further be decomposed into classes which correspond
to the system of elementary divisors:
B1 „ ppr1 | . . . |pr1 q; B2 „ p1|pr2 | . . . |pr2 q; . . . ; B̺ „ p1| . . . |1|pr̺ | . . . |pr̺ q;
B̺`1 „ p1| . . . |1|0| . . . |0q,
where the number 1 appears pν ´ 1q times in each Bν . If r1 “ r2 “ . . . “
rµ “ 0 here, then B1 , . . . , Bµ are equal to the trivial class and are therefore
left out of the decomposition; the same holds for B̺`1 if ̺ “ n. If furthermore rν “ rν`1 “ . . . “ rν`λ , then Bν`1 , . . . , Bν`λ are proper divisors
of Bν , and are therefore likewise to be excluded. Denote those remaining by Bi1 , . . . , Bik , which now provide a shortest representation, so that
Q “ rBi1 , . . . , Bik s is the unique decomposition of Q into irreducible classes.
Namely, let Bν „ p1| . . . |1|prν | . . . |prν q be representable as the least common multiple of C „ p1| . . . |1|ps1 | . . . |psλ q and D „ p1| . . . |1|pt1 | . . . |ptµ q:
then the number 1 must appear in the first pν ´ 1q positions of the system
of elementary divisors of C and D; in the νth position, the exponent s1 or
t1 must be equal to rν ; let this be s1 without loss of generality. Because
however rν “ s1 ď s2 ď sλ ď rν , it follows that C “ Bν , and so Bν is
irreducible.51 The same applies for B̺`1 , where p is replaced with 0. Each
51

In contrast, the B are reducible in one-sided classes; here no unique relationship exists
between elementary divisors and classes anymore, and therefore there is also no unique
decomposition into irreducible one-sided classes. This is shown by the following example
given to me by H. Brandt regarding the decomposition into right classes (where the classes
are represented by a basis matrix, and respectively by the corresponding module):
˜
¸
˜
¸
p 0
1 0
B “ rC1 , C2 s “ rD1 , D2 s; B „
; C1 „
;
0 p
0 p

49

of these irreducible classes gives, as the construction of the least common
multiple shows, a particular exponent, and the position where this exponent first appears in the system of elementary divisors of Q, respectively A,
while B̺`1 indicates the rank. Because these numbers are uniquely determined by the system of elementary divisors of Q, and respectively A, and
because the relation between elementary divisors and classes is bijective, the
decomposition of Q, and likewise that of an arbitrary class, into irreducible
classes is therefore unique. In summary, we can say the following: Every
two-sided class A consisting of integer matrices with a bounded number of
elements can be uniquely expressed as the least common multiple of finitely
many irreducible two-sided classes. Each irreducible class in this represents
a fixed prime divisor of the system of elementary divisors of A, an associated
exponent, and the position where this exponent first appears. The irreducible
class corresponding to the divisor 0 indicates the rank of A.
Erlangen, October 1920.

(Received on 16/10/1920.)

C2 „

˜

p
0

0
1

¸

;

D1 „

˜

1
0

0
p

¸

;

D2 „

˜

p
pp ´ 1q

0
1

¸

.

In fact, the modules pξ, pηq, ppξ, ηq, ppξ, pp ´ 1qξ ` ηq each have only the module pξ, ηq as
a proper divisor, and so are irreducible and furthermore are different from each other.

50

Translator’s notes.
Pairwise coprime (p2) - This is defined in the usual way.
Relatively prime (p2) - Although today we consider this to mean the same
as pairwise coprimality, Noether distinguishes the two.
Residual module (p25) - At the time this paper was written, the idea of a
quotient was relatively new, and so different names were in use. Lasker’s
residual module is simply another name for the same concept.
Double domain (p36) - In modern terms this is a non-commutative ring and
a module on it. Noether defines a module on a double domain to be what
in modern terms is a submodule of the module in the double domain.
Integer linear forms (p37) - Also known as integral linear combinations.
Domains (p38) - This concept does not appear to have a direct modern
equivalent. It seems to be similar to the idea of a ring, but more ambiguous
in definition. As T is clearly not itself a ring, this name has been kept.
Irreducible algebraic figure (p40) - This concept has a long history, most
notably used by Weierstrass. It is in many ways similar to the modern
notion of an algebraic variety, however there is not enough evidence to show
that they are indeed the same, so this term has been avoided.

Acknowledgements.
Special thanks go to John Rawnsley, Colin McLarty, John Stillwell, Steve
Russ, Derek Holt, Daniel Lewis and Anselm Bründlmayer for their invaluable advice, and in particular to Jeremy Gray for providing this opportunity
and for his support throughout.

51

