arXiv:1404.1901v4 [] 23 Oct 2017

INVERTIBLE IDEALS AND GAUSSIAN SEMIRINGS
SHABAN GHALANDARZADEH, PEYMAN NASEHPOUR, AND RAFIEH RAZAVI
Abstract. In the first section, we introduce the notions of fractional and
invertible ideals of semirings and characterize invertible ideals of a semidomain.
In section two, we define Prüfer semirings and characterize them in terms of
valuation semirings. In this section, we also characterize Prüfer semirings in
terms of some identities over its ideals such as (I + J)(I ∩ J) = IJ for all ideals
I, J of S. In the third section, we give a semiring version for the Gilmer-Tsang
Theorem, which states that for a suitable family of semirings, the concepts of
Prüfer and Gaussian semirings are equivalent. At last, we end this paper by
giving a plenty of examples for proper Gaussian and Prüfer semirings.

0. Introduction
Vandiver introduced the term “semi-ring” and its structure in 1934 [27], though
the early examples of semirings had appeared in the works of Dedekind in 1894,
when he had been working on the algebra of the ideals of commutative rings [4]. Despite the great efforts of some mathematicians on semiring theory in 1940s, 1950s,
and early 1960s, they were apparently not successful to draw the attention of mathematical society to consider the semiring theory as a serious line of mathematical
research. Actually, it was in the late 1960s that semiring theory was considered
a more important topic for research when real applications were found for semirings. Eilenberg and a couple of other mathematicians started developing formal
languages and automata theory systematically [6], which have strong connections
to semirings. Since then, because of the wonderful applications of semirings in engineering, many mathematicians and computer scientists have broadened the theory
of semirings and related structures [10] and [14]. As stated in [12, p. 6], multiplicative ideal theoretic methods in ring theory are certainly one of the major sources
of inspiration and problems for semiring theory. In the present paper, we develop
some ring theoretic methods of multiplicative ideal theory for semirings as follows:
Let, for the moment, R be a commutative ring with a nonzero identity. The
Dedekind-Mertens lemma in ring theory states that if f and g are two elements
of the polynomial ring R[X], then there exists a natural number n such that
c(f )n−1 c(f g) = c(f )n c(g), where by the content c(f ) of an arbitrary polynomial
f ∈ R[X], it is meant the R-ideal generated by the coefficients of f . From this, it
is clear that if R is a Prüfer domain, then R is Gaussian, i.e. c(f g) = c(f )c(g) for
all f, g ∈ R[X].
Gilmer in [9] and Tsang in [26], independently, proved that the inverse of the
above statement is also correct in this sense that if R is a Gaussian domain, then
R is a Prüfer domain.
2010 Mathematics Subject Classification. 16Y60, 13B25, 13F25, 06D75.
Key words and phrases. Semiring, Semiring polynomials, Gaussian semiring, Cancellation
ideals, Invertible ideals.
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SHABAN GHALANDARZADEH, PEYMAN NASEHPOUR, AND RAFIEH RAZAVI

Since Gaussian semirings were introduced in Definition 7 in [21] and the DedekindMertens lemma was proved for subtractive semirings in Theorem 3 in the same paper, our motivation for this work was to see how one could define invertible ideals
for semirings to use them in Dedekind-Mertens lemma and discover another family
of Gaussian semirings. We do emphasize that the definition of Gaussian semiring
used in our paper is different from the one investigated in [5] and [13]. We also
asked ourselves if some kind of a Gilmer-Tsang Theorem held for polynomial semirings. Therefore, we were not surprised to see while investigating these questions,
we needed to borrow some definitions and techniques - for example Prüfer domains
and a couple of other concepts mentioned in [18] and [8] - from multiplicative ideal
theory for rings . In most cases, we also constructed examples of proper semirings
- semirings that are not rings - satisfying the conditions of those definitions and
results to show that what we bring in this paper are really generalizations of their
ring version ones. Since different authors have used the term “semiring” with some
different meanings, it is essential, from the beginning, to clarify what we mean by
a semiring.
In this paper, by a semiring, we understand an algebraic structure, consisting of
a nonempty set S with two operations of addition and multiplication such that the
following conditions are satisfied:
(1) (S, +) is a commutative monoid with identity element 0;
(2) (S, ·) is a commutative monoid with identity element 1 6= 0;
(3) Multiplication distributes over addition, i.e. a(b + c) = ab + ac for all
a, b, c ∈ S;
(4) The element 0 is the absorbing element of the multiplication, i.e. s · 0 = 0
for all s ∈ S.
From the above definition, it is clear for the reader that the semirings are fairly
interesting generalizations of the two important and broadly studied algebraic structures, i.e. rings and bounded distributive lattices.
A nonempty subset I of a semiring S is defined to be an ideal of S if a, b ∈ I
and s ∈ S implies that a + b, sa ∈ I [3]. An ideal I of a semiring S is said to be
subtractive, if a + b ∈ I and a ∈ I implies b ∈ I for all a, b ∈ S. A semiring S is
said to be subtractive if every ideal of the semiring S is subtractive. An ideal P of
S is called a prime ideal of S if P 6= S and ab ∈ P implies that a ∈ P or b ∈ P for
all a, b ∈ S.
In §1, we define fractional and invertible ideals and show that any invertible ideal
of a local semidomain is principal (See Definitions 1.1 and 1.2 and Proposition 1.5).
Note that a semiring S is called a semidomain if for any nonzero element s of
S, sb = sc implies that b = c. A semiring is said to be local if it has only one
maximal ideal. We also prove that any invertible ideal of a weak Gaussian semilocal semidomain is principal (See Theorem 1.6). Note that a semiring is defined
to be a weak Gaussian semiring if each prime ideal of the semiring is subtractive
[21, Definition 18] and a semiring is said to be semi-local if the set of its maximal
ideals is finite. Also, note that localization of semirings has been introduced and
investigated in [15]. It is good to mention that an equivalent definition for the
localization of semirings has been given in [12, §11].
At last, in Theorem 1.8, we show that if I is a nonzero finitely generated ideal
of a semidomain S, then I is invertible if and only if Im is a principal ideal of Sm

INVERTIBLE IDEALS AND GAUSSIAN SEMIRINGS

3

for each maximal ideal m of S.
In §2, we observe that if S is a semiring, then every nonzero finitely generated
ideal of S is an invertible ideal of S if and only if every nonzero principal and every
nonzero 2-generated ideal of S is an invertible ideal of S (Check Theorem 2.1). This
result and a nice example of a proper semiring having this property, motivate us
to define Prüfer semiring, the semiring that each of its nonzero finitely generated
ideals is invertible (See Definition 2.3). After that, in Theorem 2.9, we prove that
a semidomain S is a Prüfer semiring if and only if one of the following equivalent
statements holds:
(1) I(J ∩ K) = IJ ∩ IK for all ideals I, J, and K of S,
(2) (I + J)(I ∩ J) = IJ for all ideals I and J of S,
(3) [(I + J) : K] = [I : K] + [J : K] for all ideals I, J, and K of S with K
finitely generated,
(4) [I : J] + [J : I] = S for all finitely generated ideals I and J of S,
(5) [K : I ∩ J] = [K : I] + [K : J] for all ideals I, J, and K of S with I and J
finitely generated.
Note that, in the above, it is defined that [I : J] = {s ∈ S : sJ ⊆ I}. Also, note
that this theorem is the semiring version of Theorem 6.6 in [18], though we give
partly an alternative proof for the semiring generalization of its ring version.
In §2, we also characterize Prüfer semirings in terms of valuation semirings. Let
us recall that a semidomain is valuation if its ideals are totally ordered by inclusion
[22, Theorem 2.4]. In fact, in Theorem 2.11, we prove that a semiring S is Prüfer
if and only if one of the following statements holds:
(1) For any prime ideal p of S, Sp is a valuation semidomain.
(2) For any maximal ideal m of S, Sm is a valuation semidomain.
A nonzero ideal I of a semiring S is called a cancellation ideal, if IJ = IK
implies J = K for all ideals J and K of S [17]. Let f ∈ S[X] be a polynomial over
the semiring S. The content of f , denoted by c(f ), is defined to be the S-ideal
generated by the coefficients of f . It is, then, easy to see that c(f g) ⊆ c(f )c(g)
for all f, g ∈ S[X]. Finally, a semiring S is defined to be a Gaussian semiring if
c(f g) = c(f )c(g) for all f, g ∈ S[X] [21, Definition 8].
In §3, we discuss Gaussian semirings and prove a semiring version of the GilmerTsang Theorem with the following statement (See Theorem 3.5):
Let S be a subtractive semiring such that every nonzero principal ideal of S
is invertible and ab ∈ (a2 , b2 ) for all a, b ∈ S. Then the following statements are
equivalent:
(1)
(2)
(3)
(4)

S is a Prüfer semiring,
Each nonzero finitely generated ideal of S is cancellation,
[IJ : I] = J for all ideals I and J of S,
S is a Gaussian semiring.

At last, we end this paper by giving a plenty of examples of proper Gaussian and
Prüfer semirings in Theorem 3.7 and Corollary 3.8. Actually, we prove that if S is
a Prüfer semiring (say for example S is a Prüfer domain), then FId(S) is a Prüfer
semiring, where by FId(S) we mean the semiring of finitely generated ideals of S.

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SHABAN GHALANDARZADEH, PEYMAN NASEHPOUR, AND RAFIEH RAZAVI

In this paper, all semirings are assumed to be commutative with a nonzero
identity. Unless otherwise stated, our terminology and notation will follow as closely
as possible that of [8].
1. Fractional and invertible ideals of semirings
In this section, we introduce fractional and invertible ideals for semirings and
prove a couple of interesting results for them. Note that whenever we feel it is
necessary, we recall concepts related to semiring theory to make the paper as selfcontained as possible.
Let us recall that a nonempty subset I of a semiring S is defined to be an ideal
of S if a, b ∈ I and s ∈ S implies that a + b, sa ∈ I [3]. Also, T ⊆ S is said to
be a multiplicatively closed set of S provided that if a, b ∈ T , then ab ∈ T . The
localization of S at T is defined in the following way:
First define the equivalent relation ∼ on S × T by (a, b) ∼ (c, d), if tad = tbc for
some t ∈ T . Then Put ST the set of all equivalence classes of S × T and define
addition and multiplication on ST respectively by [a, b] + [c, d] = [ad + bc, bd] and
[a, b] · [c, d] = [ac, bd], where by [a, b], also denoted by a/b, we mean the equivalence
class of (a, b). It is, then, easy to see that ST with the mentioned operations of
addition and multiplication in above is a semiring [15].
Also, note that an element s of a semiring S is said to be multiplicativelycancellable (abbreviated as MC), if sb = sc implies b = c for all b, c ∈ S. For
more on MC elements of a semiring, refer to [7]. We denote the set of all MC
elements of S by MC(S). It is clear that MC(S) is a multiplicatively closed set of
S. Similar to ring theory, total quotient semiring Q(S) of the semiring S is defined
as the localization of S at MC(S). Note that Q(S) is also an S-semimodule. For
a definition and a general discussion of semimodules, refer to [12, §14]. Now, we
define fractional ideals of a semiring as follows:
Definition 1.1. Fractional ideal. We define a fractional ideal of a semiring S to
be a subset I of the total quotient semiring Q(S) of S such that:
(1) I is an S-subsemimodule of Q(S), that is, if a, b ∈ I and s ∈ S, then a+b ∈ I
and sa ∈ I.
(2) There exists an MC element d ∈ S such that dI ⊆ S.
Let us denote the set of all nonzero fractional ideals of S by Frac(S). It is easy to
check that Frac(S) equipped with the following multiplication of fractional ideals
is a commutative monoid:
I · J = {a1 b1 + · · · + an bn : ai ∈ I, bi ∈ J}.
Definition 1.2. Invertible ideal. We define a fractional ideal I of a semiring S
to be invertible if there exists a fractional ideal J of S such that IJ = S.
Note that if a fractional ideal I of a semiring S is invertible and IJ = S, for
some fractional ideal J of S, then J is unique and we denote that by I −1 . It is
clear that the set of invertible ideals of a semiring equipped with the multiplication
of fractional ideals is an Abelian group.
Theorem 1.3. Let S be a semiring with its total quotient semiring Q(S).
(1) If I ∈ Frac(S) is invertible, then I is a finitely generated S-subsemimodule
of Q(S).

INVERTIBLE IDEALS AND GAUSSIAN SEMIRINGS

(2) If
S
(3) If
J

5

I, J ∈ Frac(S) and I ⊆ J and J is invertible, then there is an ideal K of
such that I = JK.
I ∈ Frac(S), then I is invertible if and only if there is a fractional ideal
of S such that IJ is principal and generated by an MC element of Q(S).

Proof. The proof of this theorem is nothing but the mimic of the proof of its ring
version in [18, Proposition 6.3].

Let us recall that a semiring S is defined to be a semidomain, if each nonzero
element of the semiring S is an MC element of S.
Proposition 1.4. Let S be a semiring and a ∈ S. Then the following statements
hold:
(1) The principal ideal (a) is invertible if and only if a is an MC element of S.
(2) The semiring S is a semidomain if and only if each nonzero principal ideal
of S is an invertible ideal of S.
Proof. Straightforward.



Prime and maximal ideals of a semiring are defined similar to rings ([12, §7]).
Note that the set of the unit elements of a semiring S is denoted by U (S). Also
note that when S is a semidomain, MC(S) = S − {0} and the localization of S at
MC(S) is called the semifield of fractions of the semidomain S and usually denoted
by F(S) [11, p. 22].
Proposition 1.5. Any invertible ideal of a local semidomain is principal.
Proof. Let I be an invertible ideal of a local semidomain (S, m). It is clear that
there are s1 , . . . , sn ∈ S and t1 , . . . , tn ∈ F(S), such that s1 t1 + · · · + sn tn = 1. This
implies that at least one of the elements si ti is a unit, since if all of them are nonunit,
their sum will be in m and cannot be equal to 1. Assume that s1 t1 ∈ U (S). Now
we have S = (s1 )(t1 ) ⊆ I(t1 ) ⊆ II −1 = S, which obviously implies that I = (s1 )
and the proof is complete.

Let us recall that an ideal I of a semiring S is said to be subtractive, if a + b ∈ I
and a ∈ I implies b ∈ I for all a, b ∈ S. Now we prove a similar statement for
weak Gaussian semirings introduced in [21]. Note that any prime ideal of a weak
Gaussian semiring is subtractive ([21, Theorem 19]). Using this property, we prove
the following theorem:
Theorem 1.6. Any invertible ideal of a weak Gaussian semi-local semidomain is
principal.
Proof. Let S be a weak Gaussian semi-local semidomain and Max(S) = {m1 , . . . , mn }
and II −1 = S. Similar to the proof of Proposition 1.5, for each 1 ≤ i ≤ n, there
exist ai ∈ I and bi ∈ I −1 such that ai bi ∈
/ mi . Since by [12, Corollary 7.13] any
maximal ideal of a semiring is prime, one can easily check that any mi cannot contain the intersection of the remaining maximal ideals of S. So for any 1 ≤ i ≤ n,
one can find some ui , where ui is not in mi , while it is in all the other maximal
ideals of S. Put v = u1 b1 + · · · + un bn . It is obvious that v ∈ I −1 , which causes vI
to be an ideal of S. Our claim is that vI is not a subset of any maximal ideal of
S. In contrary assume that vI is a subset of a maximal ideal, say m1 . This implies
that va1 ∈ m1 . But
va1 = (u1 b1 + · · · + un bn )a1 .

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SHABAN GHALANDARZADEH, PEYMAN NASEHPOUR, AND RAFIEH RAZAVI

Also note that ui bi ai ∈ m1 for any i ≥ 2. Since m1 is subtractive, u1 b1 a1 ∈ m1 ,
a contradiction. From all we said we have that vI = S and finally I = (v −1 ), as
required.

The proof of the following lemma is straightforward, but we bring it only for the
sake of reference.
Lemma 1.7. Let I be an invertible ideal in a semidomain S and T a multiplicatively
closed set. Then IT is an invertible ideal of ST .
Proof. Straightforward.



Let us recall that if m is a maximal ideal of S, then S − m is a multiplicatively
closed set of S and the localization of S at S − m is simply denoted by Sm [15].
Now, we prove the following theorem:
Theorem 1.8. Let I be a nonzero finitely generated ideal of a semidomain S. Then
I is invertible if and only if Im is a principal ideal of Sm for each maximal ideal m
of S.
Proof. Let S be a semidomain and I a nonzero finitely generated ideal of S.
(→) : If I is invertible, then by Lemma 1.7, Im is invertible and therefore, by
Proposition 1.5, is principal.
(←) : Assume that Im is a principal ideal of Sm for each maximal ideal m of S.
For the ideal I, define J := {x ∈ F(S) : xI ⊆ S}. It is easy to check that J is a
fractional ideal of S and IJ ⊆ S is an ideal of S. Our claim is that IJ = S. On
the contrary, suppose that IJ 6= S. So IJ lies under a maximal ideal m of S. By
hypothesis Im is principal. We can choose a generator for Im to be an element z ∈ I.
Now let a1 , . . . , an be generators of I in S. It is, then, clear that for any ai , one
can find an si ∈ S − m such that ai si ∈ (z). Set s = s1 · · · sn . Since (sz −1 )ai ∈ S,
by definition of J, we have sz −1 ∈ J. But now s = (sz −1 )z ∈ m, contradicting that
si ∈ S − m and the proof is complete.

Now the question arises if there is any proper semiring, which each of its nonzero
finitely generated ideals is invertible. The answer is affirmative and next section is
devoted to such semirings.
2. Prüfer semirings
The purpose of this section is to introduce the concept of Prüfer semirings and
investigate some of their properties. We start by proving the following important
theorem, which in its ring version can be found in [18, Theorem 6.6].
Theorem 2.1. Let S be a semiring. Then the following statements are equivalent:
(1) Each nonzero finitely generated ideal of S is an invertible ideal of S,
(2) The semiring S is a semidomain and every nonzero 2-generated ideal of S
is an invertible ideal of S.
Proof. Obviously the first assertion implies the second one. We prove that the
second assertion implies the first one. The proof is by induction. Let n > 2
be a natural number and suppose that all nonzero ideals of S generated by less
than n generators are invertible ideals and L = (a1 , a2 , . . . , an−1 , an ) be an ideal
of S. If we put I = (a1 ), J = (a2 , . . . , an−1 ) and K = (an ), then by induction’s
hypothesis the ideals I + J, J + K and K + I are all invertible ideals. On the

INVERTIBLE IDEALS AND GAUSSIAN SEMIRINGS

7

other hand, a simple calculation shows that the identity (I + J)(J + K)(K + I) =
(I + J + K)(IJ + JK + KI) holds. Also since product of fractional ideals of
S is invertible if and only if every factor of this product is invertible, the ideal
I + J + K = L is invertible and the proof is complete.

A ring R is said to be a Prüfer domain if every nonzero finitely generated ideal
of R is invertible. It is, now, natural to ask if there is any proper semiring S with
this property that every nonzero finitely generated ideal of S is invertible. In the
following remark, we give such an example.
Remark 2.2. Example of a proper semiring with this property that every nonzero
finitely generated ideal of S is invertible: Obviously (Id(Z), +, ·) is a semidomain,
since any element of Id(Z) is of the form (n) such that n is a nonnegative integer
and (a)(b) = (ab), for any a, b ≥ 0. Let I be an arbitrary ideal of Id(Z). Define AI
to be the set of all positive integers n such that (n) ∈ I and put m = min AI . Our
claim is that I is the principal ideal of Id(Z), generated by (m), i.e. I = ((m)).
For doing so, let (d) be an element of I. But then (gcd(d, m)) = (d) + (m) and
therefore, (gcd(d, m)) ∈ I. This means that m ≤ gcd(d, m), since m = min AI ,
while gcd(d, m) ≤ m and this implies that gcd(d, m) = m and so m divides d and
therefore, there exists a natural number r such that d = rm. Hence, (d) = (r)(m)
and the proof of our claim is finished. From all we said we learn that each ideal of
the semiring Id(Z) is a principal and, therefore, an invertible ideal, while obviously
it is not a ring.
By Theorem 2.1 and the example given in Remark 2.2, we are inspired to give
the following definition:
Definition 2.3. We define a semiring S to be a Prüfer semiring if every nonzero
finitely generated ideal of S is invertible.
First we prove the following interesting results:
Lemma 2.4. Let S be a Prüfer semiring. Then I ∩ (J + K) = I ∩ J + I ∩ K for
all ideals I, J, and K of S.
Proof. Let s ∈ I ∩(J +K). So there are s1 ∈ J and s2 ∈ K such that s = s1 +s2 ∈ I.
If we put L = (s1 , s2 ), by definition, we have LL−1 = S. Consequently, there are
t1 , t2 ∈ L−1 such that s1 t1 + s2 t2 = 1. So s = ss1 t1 + ss2 t2 . But st1 , st2 ∈ S, since
s = s1 + s2 ∈ L. Therefore, ss1 t1 ∈ J and ss2 t2 ∈ K. Moreover s1 t1 , s2 t2 ∈ S
and therefore, ss1 t1 , ss2 t2 ∈ I. This implies that ss1 t1 ∈ I ∩ J, ss2 t2 ∈ I ∩ K, and
s ∈ I ∩ J + I ∩ K, which means that I ∩ (J + K) ⊆ I ∩ J + I ∩ K. Since the reverse
inclusion is always true, I ∩ (J + K) = I ∩ J + I ∩ K and this finishes the proof. 
Lemma 2.5. Let S be a Prüfer semiring. Then the following statements hold:
(1) If I and K are ideals of S, with K finitely generated, and if I ⊆ K, then
there is an ideal J of S such that I = JK.
(2) If IJ = IK, where I, J and K are ideals of S and I is finitely generated
and nonzero, then J = K.
Proof. By considering Theorem 1.3, the assertion (1) holds. The assertion (2) is
straightforward.

Note that the second property in Lemma 2.5 is the concept of cancellation ideal
for semirings, introduced in [17]:

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SHABAN GHALANDARZADEH, PEYMAN NASEHPOUR, AND RAFIEH RAZAVI

Definition 2.6. A nonzero ideal I of a semiring S is called a cancellation ideal, if
IJ = IK implies J = K for all ideals J and K of S.
Remark 2.7. It is clear that each invertible ideal of a semiring is cancellation.
Also, each finitely generated nonzero ideal of a Prüfer semiring is cancellation. For
a general discussion on cancellation ideals in rings, refer to [8] and for generalizations
of this concept in module and ring theory, refer to [19] and [20].
While the topic of cancellation ideals is interesting by itself, we do not go through
them deeply. In fact in this section, we only prove the following result for cancellation ideals of semirings, since we need it in the proof of Theorem 3.5. Note that
similar to ring theory, for any ideals I and J of a semiring S, it is defined that
[I : J] = {s ∈ S : sJ ⊆ I}.
Also, we point out that this result is the semiring version of an assertion mentioned
in [8, Exercise. 4, p. 66]:
Proposition 2.8. Let S be a semiring and I be a nonzero ideal of S. Then the
following statements are equivalent:
(1) I is a cancellation ideal of S,
(2) [IJ : I] = J for any ideal J of S,
(3) IJ ⊆ IK implies J ⊆ K for all ideals J, K of S.
Proof. By considering this point that the equality [IJ : I]I = IJ holds for all ideals
I, J of S, it is then easy to see that (1) implies (2). The rest of the proof is
straightforward.

Now we prove an important theorem that is rather the semiring version of Theorem 6.6 in [18]. While some parts of our proof is similar to those ones in Theorem
6.6 in [18] and Proposition 4 in [25], other parts of the proof are apparently original.
Theorem 2.9. Let S be a semidomain. Then the following statements are equivalent:
(1) The semiring S is a Prüfer semiring,
(2) I(J ∩ K) = IJ ∩ IK for all ideals I, J, and K of S,
(3) (I + J)(I ∩ J) = IJ for all ideals I and J of S,
(4) [(I + J) : K] = [I : K] + [J : K] for all ideals I, J, and K of S with K
finitely generated,
(5) [I : J] + [J : I] = S for all finitely generated ideals I and J of S,
(6) [K : I ∩ J] = [K : I] + [K : J] for all ideals I, J, and K of S with I and J
finitely generated.
Proof. (1) P
→ (2): It is P
clear that I(J ∩ K) ⊆ IJ ∩ IK. Let s ∈ IJ ∩ IK. So we can
m
n
write s = i=1 ti zi = j=1 t′j zj′ , where ti , t′j ∈ I, zi ∈ J, and zj′ ∈ K for all 1 ≤
i ≤ m and 1 ≤ j ≤ n. Put I1 = (t1 , . . . , tm ), I2 = (t′1 , . . . , t′n ), J ′ = (z1 , . . . , zm ),
K ′ = (z1′ , . . . , zn′ ), and I3 = I1 + I2 . Then I1 J ′ ∩ I2 K ′ ⊆ I3 J ′ ∩ I3 K ′ ⊆ I3 . Since
I3 is a finitely generated ideal of S, by Lemma 2.5, there exists an ideal L of S
such that I3 J ′ ∩ I3 K ′ = I3 L. Note that L = I3 −1 (I3 J ′ ∩ I3 K ′ ) ⊆ I3 −1 (I3 J ′ ) = J ′ .
Moreover L = I3 −1 (I3 J ′ ∩ I3 K ′ ) ⊆ I3 −1 (I3 K ′ ) = K ′ . Therefore, L ⊆ J ′ ∩ K ′ . Thus
s ∈ I3 J ′ ∩ I3 K ′ = I3 L ⊆ I3 (J ′ ∩ K ′ ) ⊆ I(J ∩ K).
(2) → (3): Let I, J ⊆ S. Then (I + J)(I ∩ J) = (I + J)I ∩ (I + J)J ⊇ IJ. Since
the reverse inclusion always holds, (I + J)(I ∩ J) = IJ.

INVERTIBLE IDEALS AND GAUSSIAN SEMIRINGS

9

(3) → (1): By hypothesis, every two generated ideal I = (s1 , s2 ) is a factor of
the invertible ideal (s1 s2 ) and therefore, it is itself invertible. Now by considering
Theorem 2.1, it is clear that the semiring S is a Prüfer semiring.
(1) → (4): Let s ∈ S such that sK ⊆ I + J. So sK ⊆ (I + J) ∩ K. By Lemma
2.4, sK
Therefore, s ∈ (I ∩ K)K −1 + (J ∩ K)K −1 . Thus
Pm ⊆ I ∩ KP+n J ∩′ K.
′
s = i=1 ti zi + j=1 tj zj , where zi , zj′ ∈ K −1 , ti ∈ I ∩ K, and t′j ∈ J ∩ K for
all 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let xP∈ K and 1 ≤ i ≤ m. Then zi x, zi ti ∈
m
S and P
so ti zi x ∈ I ∩ K. Therefore, ( i=1 ti zi )K ⊆ I ∩ K ⊆ I. In a similar
n
′ ′
way, ( j=1 tj zj )K ⊆ J ∩ K ⊆ J. Thus s ∈ [I : K] + [J : K]. Therefore,
[(I + J) : K] ⊆ [I : K] + [J : K]. Since the reverse inclusion is always true,
[(I + J) : K] = [I : K] + [J : K].
(4) → (5): Let I and J be finitely generated ideals of S. Then,
S = [I + J : I + J] = [I : I + J] + [J : I + J] ⊆ [I : J] + [J : I] ⊆ S.
(5) → (6): ([25, Proposition 4]) It is clear that [K : I] + [K : J] ⊆ [K : I ∩ J].
Let s ∈ S such that s(I ∩ J) ⊆ K. By hypothesis, S = [I : J] + [J : I]. So
there exist t1 ∈ [I : J] and t2 ∈ [J : I] such that 1 = t1 + t2 . This implies
that s = st1 + st2 . Let x ∈ I. Then t2 x ∈ J. Therefore, t2 x ∈ I ∩ J. Since
s(I ∩ J) ⊆ K, st2 x ∈ K. Thus st2 ∈ [K : I]. Now let y ∈ J. Then t1 y ∈ I.
Therefore, t1 y ∈ I ∩ J. Thus st1 ∈ [K : J]. So finally we have s ∈ [K : I] + [K : J].
Therefore, [K : I ∩ J] ⊆ [K : I] + [K : J].
(6) → (1): The proof is just a mimic of the proof of [18, Theorem 6.6] and
therefore, it is omitted.

We end this section by characterizing Prüfer semirings in terms of valuation
semidomains. Note that valuation semirings have been introduced and investigated
in [22]. Let us recall that a semiring is called to be a Bézout semiring if each of its
finitely generated ideal is principal.
Proposition 2.10. A local semidomain is a valuation semidomain if and only if
it is a Bézout semidomain.
Proof. (→) : Straightforward.
(←) : Let S be a local semidomain. Take x, y ∈ S such that both of them are
nonzero. Assume that (x, y) = (d) for some nonzero d ∈ S. Define x′ = x/d and
y ′ = y/d. It is clear that there are a, b ∈ S such that ax′ + by ′ = 1. Since S is
local, one of ax′ and by ′ must be unit, say ax′ . So x′ is also unit and therefore,
(y ′ ) ⊆ S = (x′ ). Now multiplying the both sides of the inclusion by d gives us the
result (y) ⊆ (x) and by Theorem 2.4 in [22], the proof is complete.

Now we get the following nice result:
Theorem 2.11. For a semidomain S, the following statements are equivalent:
(1) S is Prüfer.
(2) For any prime ideal p, Sp is a valuation semidomain.
(3) For any maximal ideal m, Sm is a valuation semidomain.
Proof. (1) → (2) :
Let J be a finitely generated nonzero ideal in Sp , generated by s1 /u1 , . . . , sn /un ,
where si ∈ S and ui ∈ S − p. It is clear that J = Ip , where I = (s1 , . . . , sn ). By
hypothesis, I is invertible. So by Theorem 1.8, J is principal. This means that Sp

10

SHABAN GHALANDARZADEH, PEYMAN NASEHPOUR, AND RAFIEH RAZAVI

is a Bézout semidomain and since it is local, by Proposition 2.10, Sp is a valuation
semidomain.
(2) → (3) : Trivial.
(3) → (1) : Let I be a nonzero finitely generated ideal of S. Then for any
maximal ideal m of S, Im is a nonzero principal ideal of Sm and by Theorem 1.8,
I is invertible. So we have proved that the semiring S is Prüfer and the proof is
complete.

Now we pass to the next section that is on Gaussian semirings.
3. Gaussian Semirings
In this section, we discuss Gaussian semirings. For doing so, we need to recall
the concept of the content of a polynomial in semirings. Let us recall that for a
polynomial f ∈ S[X], the content of f , denoted by c(f ), is defined to be the finitely
generated ideal of S generated by the coefficients of f . In [21, Theorem 3], the
semiring version of the Dedekind-Mertens lemma (Cf. [24, p. 24] and [1]) has been
proved. We state that in the following only for the convenience of the reader:
Theorem 3.1 (Dedekind-Mertens Lemma for Semirings). Let S be a semiring.
Then the following statements are equivalent:
(1) The semiring S is subtractive, i.e. each ideal of S is subtractive,
(2) If f, g ∈ S[X] and deg(g) = m, then c(f )m+1 c(g) = c(f )m c(f g).
Now, we recall the definition of Gaussian semirings:
Definition 3.2. A semiring S is said to be Gaussian if c(f g) = c(f )c(g) for all
polynomials f, g ∈ S[X] [21, Definition 7].
Note that this is the semiring version of the concept of Gaussian rings defined
in [26]. For more on Gaussian rings, one may refer to [2] also.
Remark 3.3. There is a point for the notion of Gaussian semirings that we need
to clarify here. An Abelian semigroup G with identity, satisfying the cancellation
law, is called a Gaussian semigroup if each of its elements g, which is not a unit,
can be factorized into the product of irreducible elements, where any two such
factorizations of the element g are associated with each other [16, §8 p. 71]. In the
papers [5] and [13] on Euclidean semirings, a semiring S is called to be Gaussian if
its semigroup of nonzero elements is Gaussian, which is another notion comparing
to ours.
Finally, we emphasize that by Theorem 3.1, each ideal of a Gaussian semiring
needs to be subtractive. Such semirings are called subtractive. Note that the
boolean semiring B = {0, 1} is a subtractive semiring, but the semiring N0 is not,
since its ideal N0 − {1} is not subtractive. As a matter of fact, all subtractive ideals
of the semiring N0 are of the form kN0 for some k ∈ N0 [23, Proposition 6].
With this background, it is now easy to see that if every nonzero finitely generated ideal of a subtractive semiring S, is invertible, then S is Gaussian. Also note
that an important theorem in commutative ring theory, known as Gilmer-Tsang
Theorem (cf. [9] and [26]), states that D is a Prüfer domain if and only if D is a
Gaussian domain. The question may arise if a semiring version for Gilmer-Tsang
Theorem can be proved. This is what we are going to do in the rest of the paper.
First we prove the following interesting theorem:

INVERTIBLE IDEALS AND GAUSSIAN SEMIRINGS

11

Theorem 3.4. Let S be a semiring. Then the following statements are equivalent:
(1) S is a Gaussian semidomain and ab ∈ (a2 , b2 ) for all a, b ∈ S,
(2) S is a subtractive and Prüfer semiring.
Proof. (1) → (2): Since ab ∈ (a2 , b2 ), there exists r, s ∈ S such that ab = ra2 + sb2 .
Now define f, g ∈ S[X] by f = a + bX and g = sb + raX. It is easy to check that
f g = sab+abX +rabX 2 . Since S is Gaussian, S is subtractive by Theorem 3.1, and
we have c(f g) = c(f )c(g), i.e. (ab) = (a, b)(sb, ra). But (ab) = (a)(b) is invertible
and therefore, (a, b) is also invertible and by Theorem 2.1, S is a Prüfer semiring.
(2) → (1): Since S is a subtractive and Prüfer semiring, by Theorem 3.1, S is a
Gaussian semiring. On the other hand, one can verify that (ab)(a, b) ⊆ (a2 , b2 )(a, b)
for any a, b ∈ S. If a = b = 0, then there is nothing to be proved. Otherwise, since
(a, b) is an invertible ideal of S, we have ab ∈ (a2 , b2 ) and this completes the
proof.

Theorem 3.5 (Gilmer-Tsang Theorem for Semirings). Let S be a subtractive
semidomain such that ab ∈ (a2 , b2 ) for all a, b ∈ S. Then the following statements
are equivalent:
(1) S is a Prüfer semiring,
(2) Each nonzero finitely generated ideal of S is cancellation,
(3) [IJ : I] = J for all finitely generated ideals I and J of S,
(4) S is a Gaussian semiring.
Proof. Obviously (1) → (2) and (2) → (3) hold by Proposition 2.5 and Proposition
2.8, respectively.
(3) → (4): Let f, g ∈ S[X]. By Theorem 3.1, we have c(f )c(g)c(f )m =
c(f g)c(f )m . So [c(f )c(g)c(f )m : c(f )m ] = [c(f g)c(f )m : c(f )m ]. This means that
c(f )c(g) = c(f g) and S is Gaussian.
Finally, the implication (4) → (1) holds by Theorem 3.4 and this finishes the
proof.

Remark 3.6. In [21, Theorem 9], it has been proved that every bounded distributive lattice is a Gaussian semiring. Also, note that if L is a bounded distributive
lattice with more than two elements, it is neither a ring nor a semidomain, since if
it is a ring then the idempotency of addition causes L = {0} and if it is a semidomain, the idempotency of multiplication causes L = B = {0, 1}. With the help
of the following theorem, we give a plenty of examples of proper Gaussian and
Prüfer semirings. Let us recall that if S is a semiring, then by FId(S), we mean
the semiring of finitely generated ideals of S.
Theorem 3.7. Let S be a Prüfer semiring. Then the following statements hold for
the semiring FId(S):
(1) FId(S) is a Gaussian semiring.
(2) FId(S) is a subtractive semiring.
(3) FId(S) is an additively idempotent semidomain and for all finitely generated
ideals I and J of S, we have IJ ∈ (I 2 , J 2 ).
(4) FId(S) is a Prüfer semiring.
Proof. (1): Let I, J ∈ FId(S). Since S is a Prüfer semiring and I ⊆ I + J, by
Theorem 1.3, there exists an ideal K of S such that I = K(I + J). On the other
hand, since I is invertible, K is also invertible. This means that K is finitely

12

SHABAN GHALANDARZADEH, PEYMAN NASEHPOUR, AND RAFIEH RAZAVI

generated and therefore, K ∈ FId(S) and I ∈ (I + J). Similarly, it can be proved
that J ∈ (I + J). So, we have (I, J) = (I + J) and by [21, Theorem 8], FId(S) is a
Gaussian semiring.
(2): By Theorem 3.1, every Gaussian semiring is subtractive. But by (1), FId(S)
is a Gaussian semiring. Therefore, FId(S) is a subtractive semiring.
(3): Obviously FId(S) is additively-idempotent and since S is a Prüfer semiring,
FId(S) is an additively idempotent semidomain. By Theorem [12, Proposition 4.43],
we have (I + J)2 = I 2 + J 2 and so (I + J)2 ∈ (I 2 , J 2 ). But (I + J)2 = I 2 + J 2 + IJ
and by (2), FId(S) is subtractive. So, IJ ∈ (I 2 , J 2 ), for all I, J ∈ FId(S).
(4): Since FId(S) is a Gaussian semidomain such that IJ ∈ (I 2 , J 2 ) for all
I, J ∈ FId(S), by Theorem 3.4, FId(S) is a Prüfer semiring and this is what we
wanted to prove.

Corollary 3.8. If D is a Prüfer domain, then FId(D) is a Gaussian and Prüfer
semiring.
Acknowledgments
The authors are very grateful to the anonymous referee for her/his useful pieces
of advice, which helped them to improve the paper. The first named author is supported by the Faculty of Mathematics at the K. N. Toosi University of Technology.
The second named author is supported by Department of Engineering Science at
University of Tehran and Department of Engineering Science at Golpayegan University of Technology and his special thanks go to the both departments for providing
all necessary facilities available to him for successfully conducting this research.
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Shaban Ghalandarzadeh, Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
E-mail address: ghalandarzadeh@kntu.ac.ir
Peyman Nasehpour, Department of Engineering Science, Golpayegan University of
Technology, Golpayegan, Iran
Peyman Nasehpour, Department of Engineering Science, Faculty of Engineering,
University of Tehran, Tehran, Iran
E-mail address: nasehpour@gmail.com, nasehpour@gut.ac.ir
Rafieh Razavi, Faculty of Mathematics, K. N. Toosi University of Technology,
Tehran, Iran
E-mail address: rrazavi@mail.kntu.ac.ir

