GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH
ZERO-DIVISORS

arXiv:1312.7403v1 [] 28 Dec 2013

CHRISTOPHER PARK MOONEY
Abstract. Recently substantial progress has been made on generalized factorization techniques in integral domains, in particular τ -factorization. There has also been advances made
in investigating factorization in commutative rings with zero-divisors. One approach which
has been found to be very successful is that of U-factorization introduced by C.R. Fletcher.
We seek to synthesize work done in these two areas by generalizing τ -factorization to rings
with zero-divisors by using the notion of U-factorization.
2010 AMS Subject Classification: 13A05, 13E99, 13F15

1. Introduction
Much work has been done on generalized factorization techniques in integral domains.
There is an excellent overview in [4], where particular attention is paid to τ -factorization.
Several authors have investigated ways to extend factorization to commutative rings with
zero-divisors. For instance, D.D. Anderson, Valdez-Leon, Aǧargün, Chun [3, 6, 7]. One
particular method was that of U-factorization introduced by C.R. Fletcher in [11] and [12].
This method of factorization has been studied extensively by Michael Axtell and others
in [9, 10, 8]. We synthesize this work done into a single study of what we will call τ -Ufactorization.
In this paper, we will assume R is a commutative ring with 1. Let R∗ = R − {0}, let
U(R) be the set of units of R, and let R# = R∗ − U(R) be the non-zero, non-units of
R. As in [10], we define U-factorization as follows. Let a ∈ R be a non-unit. If a =
λa1 · · · an b1 · · · bm is a factorization with λ ∈ U(R), ai , bi ∈ R# , then we will call a =
λa1 a2 · · · an ⌈b1 b2 · · · bm ⌉ a U-factorization of a if (1) ai (b1 · · · bm ) = (b1 · · · bm ) for all 1 ≤ i ≤
n and (2) bj (b1 · · · bbj · · · bm ) 6= (b1 · · · bbj · · · bm ) for 1 ≤ j ≤ m where bbj means bj is omitted
from the product. Here (b1 · · · bm ) is the principal ideal generated by b1 · · · bm . The bi ’s in
this particular U-factorization above will be referred to as essential divisors. The ai ’s in this
particular U-factorization above will be referred to as inessential divisors. A U-factorization
is said to be trivial if there is only one essential divisor.
Note: we have added a single unit factor in front with the inessential divisors which was
not in M. Axtell’s original paper. This is added for consistency with the τ -factorization
definitions and it is evident that a unit is always inessential. We allow only one unit factor,
so it will not affect any of the finite factorization properties.
Remark. If a = λa1 · · · an ⌈b1 · · · bm ⌉ is a U-factorization, then for any 1 ≤ i0 ≤ m, we have
(a) = (b1 · · · bm ) ( (b1 · · · bc
i0 · · · bm ). This is immediate from the definition of U-factorization.
Date: November 6, 2017.
Key words and phrases. factorization, zero-divisors, commutative rings.
1

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CHRISTOPHER PARK MOONEY

In [9], M. Axtell defines a non-unit a and b to be associate if (a) = (b) and a non-zero nonunit a said to be irreducible if a = bc implies a is associate to b or c. R is commutative ring
R to be U-atomic if every non-zero non-unit has a U-factorization in which every essential
divisor is irreducible. R is said to be a U-finite factorization ring if every non-zero non-unit
has a finite number of distinct U-factorizations. R is said to be a U-bounded factorization
ring if every non-zero non-unit has a bound on the number of essential divisors in any Ufactorization. R is said to be a U-weak finite factorization ring if every non-zero non-unit
has a finite number of non-associate essential divisors. R is said to be a U-atomic idf-ring if
every non-zero non-unit has a finite number of non-associate irreducible essential divisors. R
is said to be a U-half factorization ring if R is U-atomic and every U-atomic factorization has
the same number of irreducible essential divisors. R is said to be a U-unique factorization
ring if it is a U-HFR and in addition each U-atomic factorization can be arranged so the
essential divisors correspond up to associate. In [10, Theorem 2.1], it is shown this definition
of U-UFR is equivalent to the one given by C.R. Fletcher in [11, 12].
In the second section, we begin with some preliminary definitions and results about τ factorization in integral domains as well as factorization in rings with zero-divisors. In the
third section, we state definitions for τ -U-irreducible elements and τ -U-finite factorization
properties. We also prove some preliminary results using these new definitions. In the
fourth section, we demonstrate the relationship between rings satisfying the various τ -U finite
factorization properties. Furthermore, we compare these properties with the rings satisfying
τ -finite factorization properties studied in [13]. In the final section, we investigate direct
products of rings. We introduce a relation τ× which carries many τ -U-finite factorization
properties of the component rings through the direct product.
2. Preliminary Definitions and Results
As in [6], we let a ∼ b if (a) = (b), a ≈ b if there exists λ ∈ U(R) such that a = λb, and
a∼
= b if (1) a ∼ b and (2) a = b = 0 or if a = rb for some r ∈ R then r ∈ U(R). We say a
and b are associates (resp. strong associates, very strong associates) if a ∼ b (resp. a ≈ b,
a∼
= b). As in [1], a ring R is said to be strongly associate (resp. very strongly associate) ring
if for any a, b ∈ R, a ∼ b implies a ≈ b (resp. a ∼
= b).
Let τ be a relation on R# , that is, τ ⊆ R# × R# . We will always assume further that τ is
symmetric. Let a be a non-unit, ai ∈ R# and λ ∈ U(R), then a = λa1 · · · an is said to be a
τ -factorization if ai τ aj for all i 6= j. If n = 1, then this is said to be a trivial τ -factorization.
Each ai is said to be a τ -factor, or that ai τ -divides a, written ai |τ a.
We say that τ is multiplicative (resp. divisive) if for a, b, c ∈ R# (resp. a, b, b′ ∈ R# ), aτ b
and aτ c imply aτ bc (resp. aτ b and b′ | b imply aτ b′ ). We say τ is associate (resp. strongly
associate, very strongly associate) preserving if for a, b, b′ ∈ R# with b ∼ b′ (resp. b ≈ b′ ,
b ∼
= b′ ) aτ b implies aτ b′ . We define a τ -refinement of a τ -factorization λa1 · · · an to be a
factorization of the form
(λλ1 · · · λn ) · b11 · · · b1m1 · b21 · · · b2m2 · · · bn1 · · · bnmn
where ai = λi bi1 · · · bimi is a τ -factorization for each i. This is slightly different from the
original definition in [4] where no unit factor was allowed, and one can see they are equivalent
when τ is associate preserving. We then say that τ is refinable if every τ -refinement of a
τ -factorization is a τ -factorization. We say τ is combinable if whenever λa1 · · · an is a τ factorization, then so is each λa1 · · · ai−1 (ai ai+1 )ai+2 · · · an .

GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

3

We now summarize several of the definitions given in [13]. Let a ∈ R be a non-unit.
Then a is said to be τ -irreducible or τ -atomic if for any τ -factorization a = λa1 · · · an , we
have a ∼ ai for some i. We will say a is τ -strongly irreducible or τ -strongly atomic if for
any τ -factorization a = λa1 · · · an , we have a ≈ ai for some ai . We will say that a is τ -mirreducible or τ -m-atomic if for any τ -factorization a = λa1 · · · an , we have a ∼ ai for all i.
Note: the m is for “maximal” since such an a is maximal among principal ideals generated
by elements which occur as τ -factors of a. We will say that a is τ -very strongly irreducible or
τ -very strongly atomic if a ∼
= a and a has no non-trivial τ -factorizations. See [13] for more
equivalent definitions of these various forms of τ -irreducibility.
From [13, Theorem 3.9], we have the following relations where † represents the implication
requires a strongly associate ring:
+3 τ -strongly irred.
+3 τ -irred.
τ -very strongly irred.
KS
❯❯❯❯
♠♠ 2:
♠
♠
❯❯❯❯❯
♠♠
†
❯❯❯❯❯
♠♠♠
❯ &.
♠♠♠
τ -m-irred.

3. τ -U-irreducible elements
A τ -U-factorization of a non-unit a ∈ R is a U-factorization a = λa1 a2 · · · an ⌈b1 b2 · · · bm ⌉
for which λa1 · · · an b1 · · · bm is also a τ -factorization.
Given a symmetric relation τ on R# , we say R is τ -U-refinable if for every τ -U-factorization
of any non-unit a ∈ U(R), a = λa1 · · · an ⌈b1 · · · bm ⌉, any τ -U-factorization of an essential
divisors, bi = λ′ c1 · · · cn′ ⌈d1 · · · dm′ ⌉ satisfies
a = λλ′ a1 · · · an c1 · · · cn′ ⌈b1 · · · bi−1 d1 · · · dm′ bi+1 · · · bm ⌉
is a τ -U-factorization.
Example 3.1. Let R = Z/20Z, and let τ = R# × R# .
Certainly 0 = ⌈10 · 10⌉ is a τ -U-factorization. But 10 = ⌈2 · 5⌉ is a τ -U-factorization;
however, 0 = ⌈2 · 5 · 2 · 6⌉ is not a U-factorization since 5 becomes inessential after a τ U-refinement. It will sometimes be important to ensure the essential divisors of a τ -Urefinement of a τ -U-factorization’s essential divisors remain essential. We will see that in
a présimplifiable ring, there are no inessential divisors, so for τ -refinable, R will be τ -Urefinable.
As stated in [9], the primary benefit of looking at U-factorizations is the elimination of
troublesome idempotent elements that ruin many of the finite factorization properties. For
instance, even Z6 is not a BFR (a ring in which every non-unit has a bound on the number of
non-unit factors in any factorization) because we have 3 = 32 . Thus, 3 is an idempotent, so
3 = 3n for all n ≥ 1 which yields arbitrarily long factorizations. When we use U-factorization,
we see any of these factorizations can be rearranged to 3 = 3n−1 ⌈3⌉, which has only one
essential divisor.
Let α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible}. Let a
be a non-unit. If a = λa1 a2 · · · an ⌈b1 b2 · · · bm ⌉ is a τ -U-factorization, then this factorization
is said to be a τ -U-α-factorization if it is a τ -U-factorization and the essential divisors bi are
τ -α for 1 ≤ i ≤ m.
One must be somewhat more careful with U-factorizations as there is a loss of uniqueness

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CHRISTOPHER PARK MOONEY

in the factorizations. For instance, if we let R = Z6 × Z8 , then we can factor (3, 4) as
(3, 1) ⌈(3, 3)(1, 4)⌉ or (3, 3) ⌈(3, 1)(1, 4)⌉. On the bright side, we have [8, Proposition 4.1].
Theorem 3.2. Every factorization can be rearranged into a U-factorization.
Corollary 3.3. Let R be a commutative ring with 1 and τ a symmetric relation on R# .
Let α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible}. For every
τ -α factorization of a non-unit a ∈ R, a = λa1 · · · an , we can rearrange this factorization
into a τ -U-α-factorization.
Proof. Let a = λa1 · · · an be a τ -α-factorization. By Theorem 3.2 we can rearrange this to
form a U-factorization. This remains a τ -factorization since τ is assumed to be symmetric.
Lastly each ai is τ -α, so the essential divisors are τ -α.

This leads us to another equivalent definition of τ -irreducible.
Theorem 3.4. Let a ∈ R be a non-unit. Then a is τ -irreducible if and only if any τ -Ufactorization of a has only one essential divisor.
Proof. (⇒) Let a be τ -irreducible. Let a = λa1 · · · an ⌈b1 · · · bm ⌉ be a τ -U-factorization.
Suppose m ≥ 2, then a = λa1 · · · an b1 · · · bm is a τ -factorization implies a ∼ ai0 for some
1 ≤ i0 ≤ n or a ∼ bi0 for some 1 ≤ i0 ≤ m. But then either

or

(a) = (a1 · · · an b1 · · · bm ) ( (a1 · · · an bb1 b2 · · · bm ) ⊆ (ai0 ) = (a)

d
c
(a) = (a1 · · · an b1 · · · bm ) = (b1 · · · bm ) ( (bb1 · · · bd
i0 −1 · bi0 · bi0 +1 · · · bm ) ⊆ (bi0 ) = (a)

a contradiction.
(⇐) Suppose a = λa1 · · · an . Then this can be rearranged into a U-factorization, and
hence a τ -U-factorization. By hypothesis, there can only be one essential divisor. Suppose
it is an . We have a = λa1 · · · an−1 ⌈an ⌉ is a τ -U-factorization and a ∼ an as desired.

We now define the finite factorization properties using the τ -U-factorization approach.
Let α ∈ { irreducible, strongly irreducible, m-irreducible, very strongly irreducible } and
let β ∈ {associate, strongly associate, very strongly associate }. R is said to be τ -U-α if
for all non-units a ∈ R, there is a τ -U-α-factorization of a. R is said to satisfy τ -U-ACCP
(ascending chain condition on principal ideals) if every properly ascending chain of principal
ideals (a1 ) ( (a2 ) ( · · · such that ai+1 is an essential divisor in some τ -U-factorization of ai ,
for each i terminates after finitely many principal ideals. R is said to be a τ -U-BFR if for all
non-units a ∈ R, there is a bound on the number of essential divisors in any τ -U-factorization
of a.
R is said to be a τ -U-β-FFR if for all non-units a ∈ R, there are only finitely many
τ -U-factorizations up to rearrangement of the essential divisors and β. R is said to be a τ U-β-WFFR if for all non-units a ∈ R, there are only finitely many essential divisors among
all τ -U-factorizations of a up to β. R is said to be a τ -U-α-β-divisor finite (df ) ring if
for all non-units a ∈ R, there are only finitely many essential τ -α divisors up to β in the
τ -U-factorizations of a.
R is said to be a τ -U-α-HFR if R is τ -U-α and for all non-units a ∈ R, the number of
essential divisors in any τ -U-α-factorization of a is the same. R is said to be a τ -U-α-βUFR if R is a τ -U-α-HFR and the essential divisors of any two τ -U-α-factorizations can be

GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

5

rearranged to match up to β.
R is said to be présimplifiable if for every x ∈ R, x = xy implies x = 0 or y ∈ U(R).
This is a condition which has been well studied and is satisfied by any domain or local ring.
We introduce two slight modifications of this. R is said to be τ -présimplifiable if for every
x ∈ R, the only τ -factorizations of x which contain x as a τ -factor are of the form x = λx
for a unit λ. R is said to be τ -U-présimplifiable if for every non-zero non-unit x ∈ R, all
τ -U-factorizations have no non-unit inessential divisors.
Theorem 3.5. Let R be a commutative ring with 1 and let τ be a symmetric relation on
R# . We have the following.
(1) If R is présimplifiable, then R is τ -U-présimplifiable.
(2) If R is τ -U-présimplifiable, then R is R is τ -présimplifiable.
That is présimplifiable ⇒ τ -U-présimplifiable ⇒ τ -présimplifiable. If τ = R# × R# , then all
are equivalent.
Proof. (1) Let R be présimplifiable, and x ∈ R# . Suppose x = λa1 · · · an ⌈b1 · · · bm ⌉ is a
τ -U-factorization. Then (x) = (b1 · · · bm ). R présimplifiable implies that all the associate
relations coincide, so in fact x ∼
= b1 · · · bm implies that λa1 · · · an ∈ U(R) and hence all
inessential divisors are units.
(2) Let R be τ -U-présimplifiable, and x ∈ R such that x = λxa1 · · · an is a τ -factorization.
We claim that x = λa1 · · · an ⌈x⌉ is a τ -U-factorization. For any 1 ≤ i ≤ n, x | ai x and
(ai x)(λa1 · · · abi · · · an ) = x shows ai x | x, proving the claim. This implies λa1 · · · an ∈ U(R)
as desired.
Let τ = R# × R# and suppose R is τ -présimplifiable. Suppose x = xy, for x 6= 0, we show
y ∈ U(R). If x ∈ U(R), then multiplying through by x−1 yields 1 = x−1 x = x−1 xy = y and
y ∈ U(R) as desired. We may now assume x ∈ R# . If y = 0, then x = 0, a contradiction.
If y ∈ U(R) we are already done, so we may assume y ∈ R# . Thus xτ y, and x = xy is a
τ -factorization, so y ∈ U(R) as desired.


4. τ -U-finite factorization relations
We now would like to show the relationship between rings with various τ -U-α-finite factorization properties as well as compare these rings with the τ -α-finite factorization properties
of [13].
Theorem 4.1. Let R be a commutative ring with 1 and let τ be a symmetric relation on
R# . Consider the following statements.
(1) R is a τ -BFR.
(2) R is τ -présimplifiable and for every non-unit a1 ∈ R, there is a fixed bound on the length
of chains of principal ideals (ai ) ascending from a1 such that at each stage ai+1 |τ ai .
(3) R is τ -présimplifiable and a τ -U-BFR.
(4) For every non-unit a ∈ R , there are natural numbers N1 (a) and N2 (a) such that if
a = λa1 · · · an ⌈b1 · · · bm ⌉ is a τ -U-factorization, then n ≤ N1 (a) and m ≤ N2 (a).
Then (4) ⇒ (1) and (2) ⇒ (3). For τ refinable, (1) ⇒ (2) and for R τ -U-présimplifiable,
(3) ⇒ (4). Thus all are equivalent if R is τ -U-présimplifiable and τ is refinable.
Let ⋆ represent τ being refinable, and † represent R being τ -U-présimplifiable, then the

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CHRISTOPHER PARK MOONEY

following diagram summarizes the theorem.
(1)

⋆

KS

(4) ks

+3

(2)


†

(3)

Proof. (1) ⇒ (2) Let τ be refinable. Suppose there were a non-trivial τ -factorization x =
λxa1 · · · an with n ≥ 1. Since τ is assumed to be refinable we can continue to replace the
τ -factor x with this factorization.
x = λxa1 · · · an = (λλ)xa1 · · · an a1 · · · an = · · · = (λλλ)xa1 · · · an a1 · · · an a1 · · · an = · · ·
yields an unbounded series of τ -factorizations of increasing length.
Let a1 be a non-unit in R. Suppose N is the bound on the length of any τ -factorization
of a1 . We claim that N satisfies the requirement of (2). Let (a1 ) ( (a2 ) ( · · · be an
ascending chain of principal ideals generated by elements which satisfy ai+1 |τ ai for each i.
Say ai = λi ai+1 ai1 · · · aini for each i. Furthermore, we can assume ni ≥ 1 for each i or else
the containment would not be proper. Then we can write
a1 = λ1 a2 a11 · · · a1n1 = λ1 λ2 a3 a21 · · · a2n2 a11 · · · a1n1 = · · · .
Each remains a τ -factorization since τ is refinable and we have added at least one factor
at each step. If the chain were greater than length N we would contradict R being a τ -BFR.
(2) ⇒ (3) Let a ∈ R be a non-unit. Let N be the bound on the length of any properly ascending chain of principle ideals ascending from a such that ai+1 |τ ai . If a =
λa1 · · · an ⌈b1 · · · bm ⌉ is a τ -U-factorization, then we get an ascending chain with b1 · · · bi−1 |τ
b1 · · · bi for each i:
(a) = (b1 · · · bm ) ( (b1 · · · bm−1 ) ( (b1 · · · bm−2 ) ( · · · ( (b1 b2 ) ( (b1 ).
Hence, m ≤ N and we have found a bound on the number of essential divisors in any τ -Ufactorization of a, making R a τ -U-BFR.
(3) ⇒ (4) Let a ∈ R be a non-unit. Let Ne (a) be the bound on the number of essential
divisors in any τ -U-factorization of a. Since R is τ -U-présimplifiable, there are no inessential
τ -U-divisors of a. We can set N1 (a) = 0, and N2 (a) = Ne (a) and see that this satisfies the
requirements of the theorem.
(4) ⇒ (1) Let a ∈ R be a non-unit. Then any τ -factorization
 a = λa1 · · · an can be

rearranged into a τ -U-factorization, say a = λas1 · · · asi asi+1 · · · asn . But then n = i + (n −
i) ≤ N1 (a) + N2 (a). Hence the length of any τ -factorization must be less than N1 (a) + N2 (a)
proving R is a τ -BFR as desired.

The way we have defined our finite factorization properties on only the essential divisors
causes a slight problem. Given a τ -U-factorization a = λa1 · · · an ⌈b1 · · · bm ⌉, we only know
that a ∼ b1 · · · bm . This may no longer be a τ -factorization of a, but rather only some
associate of a. This is easily remedied by insisting that our rings are strongly associate.
Lemma 4.2. Let R be a strongly associate ring with τ a symmetric relation on R# , and let
α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible}. Let a ∈ R,
a non-unit. If a = λa1 a2 · · · an ⌈b1 b2 · · · bm ⌉ is a τ -U-α-factorization, then there is a unit
µ ∈ U(R) such that a = µb1 · · · bm is a τ -α-factorization.

GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

7

Proof. Let a = λa1 a2 · · · an ⌈b1 b2 · · · bm ⌉ be a τ -U-α-factorization. By definition, (a) =
(b1 · · · bm ), and R strongly associate implies that a ≈ b1 · · · bm . Let µ ∈ U(R) be such
that a = µb1 · · · bm . We still have bi τ bj for all i 6= j, and bi is τ -α for every i. Hence
a = µb1 · · · bm is the desired τ -factorization, proving the lemma.

Theorem 4.3. Let R be a commutative ring with 1, and let τ be a symmetric relation on
R# . Let α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible}, and
β ∈ {associate, strongly associate, very strongly associate }. We have the following.
(1) If R is τ -α, then R is τ -U-α.
(2) If R satisfies τ -ACCP, then R satisfies τ -U-ACCP.
(3) If R is a τ -BFR, then R is a τ -U-BFR.
(4) If R is a τ -β-FFR, then R is a τ -U-β-FFR.
(5) Let R be a τ -β-WFFR, then R is a τ -U-β-WFFR.
(6) Let R be a τ -α-β-divisor finite ring, then R is τ -U-α-β-divisor finite ring.
(7) Let R be a strongly associate τ -α-HFR (resp. τ -α-β-UFR), then R is τ -U-α-HFR (resp.
τ -U-α-β-UFR).
Proof. (1) This is immediate from Corollary 3.3.
(2) Suppose there were a infinite properly ascending chain of principal ideals (a1 ) ( (a2 ) (
· · · such that ai+1 is an essential divisor in some τ -U-factorization of ai , for each i. Every
essential τ -U-divisor is certainly a τ -divisor. This would contradict the fact that R satisfies
τ -ACCP.
(3) We suppose that there is a non-unit a ∈ R with τ -U-factorizations having arbitrarily
large numbers of essential τ -U-divisors. Each is certainly a τ -factorization, having at least
as many τ -factors as there are essential τ -divisors, so this would contradict the hypothesis.
(4) Every τ -U-factorization is certainly among the τ -factorizations. If the latter is finite,
then so is the former.
(5) For any given non-unit a ∈ R, every essential τ -U-divisor of a is certainly a τ -factor
of a which has only finitely many up to β. Hence there can be only finitely many essential
τ -U-factors up to β.
(6) Let a ∈ R be a non-unit. Every essential τ -U-α-divisor of a is a τ -α-factor of a.
There are only finitely many τ -α-divisors up to β, so then there can be only finitely many
τ -U-α-divisors of a up to β.
(7) We have already seen that R being τ -α implies R is τ -U-α. Let a ∈ R be a non-unit.
We suppose for a moment there are two τ -α-U-factorizations:
a = λa1 · · · an ⌈b1 · · · bm ⌉ = λ′ a′1 · · · a′n′ ⌈b′1 · · · b′m′ ⌉
such that m 6= m′ (resp. m 6= m′ or there is no rearrangement such that bi and b′i are β
for each i). Lemma 4.2 implies ∃µ, µ′ ∈ U(R) with a = µb1 · · · bm = µ′ b′1 · · · b′m′ are two
τ -α-factorizations of a, so m = m′ (resp. m = m′ and there is a rearrangement so that bi
and b′i are β for each 1 ≤ i ≤ m), a contradiction, proving R is indeed a τ -U-α-HFR (resp.
-β-UFR) as desired.

Theorem 4.4. Let R be a commutative ring with 1 and τ a symmetric relation on R# .
Let α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible}, and let
β ∈ {associate, strongly associate, very strongly associate}.
(1) If R is a τ -U-α-β-UFR, then R is a τ -α-U-HFR.
(2) If R is τ -U-refinable and R is a τ -U-α-β-UFR, then R is a τ -U-β-FFR.

8

(3)
(4)
(5)
(6)
(7)
(8)

CHRISTOPHER PARK MOONEY

If
If
If
If
If
If

R
R
R
R
R
R

is
is
is
is
is
is

τ -U-refinable and R is a τ -U-α-HFR, then R is a τ -U-BFR.
a τ -U-β-FFR, then R is a τ -U-BFR.
a τ -U-β-FFR, then R is a τ -U-β-WFFR.
a τ -U-β-WFFR, then R is a τ -U-α-β-divisor finite ring.
τ -U-refinable and R is a τ -U-α-BFR, then R satisfies τ -U-ACCP.
τ -U-refinable and R satisfies τ -U-ACCP, then R is τ -U-α.

Proof. (1) This is immediate from definitions.
(2) Let a ∈ R be a non-unit. Let a = λa1 · · · an ⌈b1 · · · bm ⌉ be the unique τ -α-U-factorization
up to rearrangement and β. Given any other τ -U-factorization, we can τ -U-refine each essential τ -U-divisor into a τ -U-α-factorization of a. There is a rearrangement of the essential
divisors to match up to β with bi for each 1 ≤ i ≤ m. Thus the essential divisors in any τ -Ufactorization come from some combination of products of β of the m τ -U-α essential factors
in our original factorization. Hence there are at most 2m possible distinct τ -U-factorizations
up to β, making this a τ -U-β-FFR as desired.
(3) For a given non-unit a ∈ R, the number of essential divisors in any τ -U-α-factorization
is the same, say N. We claim this is a bound on the number of essential divisors of any
τ -U-factorization. Suppose there were a τ -U-factorization a = λa1 · · · an ⌈b1 · · · bm ⌉ with
m > N. For every i, bi has a τ -U-α-factorization with at least one essential divisor. Since R
is τ -U-refinable, we can τ -U-refine the factorization yielding a τ -U-α-factorization of a with
at least m τ -U-α essential factors. This contradicts the assumption that R is a τ -U-α-HFR.
(4) Let R be a τ -U-β-FFR. Let a ∈ R be a non-unit. There are only finitely many τ -Ufactorizations of a up to rearrangement and β of the essential divisors. We can simply take
the maximum of the number of essential divisors among all of these factorizations. This is
an upper bound for the number of essential divisors in any τ -U-factorization.
(5) Let R be a τ -U-β-FFR, then for any non-unit a ∈ R. Let S be the collection of essential
divisors in the finite number of representative τ -U-factorizations of a up to β. This gives us
a finite collection of elements up to β. Every essential divisor up to β in a τ -U-factorization
of a must be among these, so this collection is finite as desired.
(6) If every non-unit a ∈ R has a finite number of proper essential τ -U divisors, then
certainly there are a finite number of essential τ -α-U-divisors.
(7) Suppose R is a τ -U-BFR, but (a1 ) ( (a2 ) ( · · · is a properly ascending chain of
principal ideals such that ai+1 is an essential factor in some τ -U-factorization of ai , say
ai = λi ai1 · · · aini ⌈ai+1 bi1 · · · bimi ⌉
for each i. Furthermore, mi ≥ 1, for each i otherwise we would have (ai+1 ) = (ai ) contrary
to our assumption that our chain is properly increasing. Our assumption that R is τ -U
refinable allows us to factor a1 as follows:
a1 = λ1 a11 · · · a1n1 ⌈a2 b11 · · · b1m1 ⌉ =
λ1 λ2 a11 · · · a1n1 a21 · · · a2n2 ⌈a3 b21 · · · b2m2 b11 · · · b1m1 ⌉
and so on. At each iteration i we have at least i + 1 essential factors in our τ -U-factorization.
This contradicts the assumption that a1 should have a bound on the number of essential
divisors in any τ -U-factorization.
(8) Let a1 ∈ R be a non-unit. If a1 is τ -U-α we are already done, so there must be a
non-trivial τ -U factorization of a1 , say:
a1 = λ1 a11 · · · a1n1 ⌈a2 b11 · · · b1m1 ⌉ .

GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

9

Now if all of the essential divisors are τ -U-α we are done as we have found a τ -U-αfactorization. After rearranging if necessary, we suppose that a2 is not τ -U-α. Therefore a2
has a non-trivial τ -U-factorization, say:
a2 = λ2 a21 · · · a2n1 ⌈a3 b21 · · · b2m2 ⌉ .
Because R is τ -U-refinable, this gives us a τ -U-factorization:
a1 = λ1 λ2 a11 · · · a1n1 a21 · · · a2n2 ⌈a3 b21 · · · b2m2 b11 · · · b1m1 ⌉
which cannot be τ -U-α or else we would be done. We can continue in this fashion and get
an ascending chain of principal ideals
(a1 ) ⊆ (a2 ) ⊆ · · ·
such that ai+1 is an essential τ -U-divisor of ai for each i.
Claim: This chain must be properly ascending. Suppose (ai ) = (ai+1 ) for some i. When
we look at ai = λi ai1 · · · aini ⌈ai+1 bi1 · · · bimi ⌉, we see that (ai ) = (ai+1 bi1 · · · bimi ). But then
we could remove any of the bij for any 1 ≤ j ≤ mi and still have (ai ) = (ai+1 bi1 · · · bc
ij · · · bimi )
contradicting the fact that the factorization was a τ -U-factorization since bij is inessential.
We certainly have (ai ) ⊆ (ai+1 bi1 · · · bc
ij · · · bimi ). To see the other containment holds,
(ai ) = (ai+1 ) ⇒ ai+1 = ai r for some r ∈ R, and we can simply multiply by bi1 · · · bc
ij · · · bimi
on both sides to see that
c
ai+1 bi1 · · · bc
ij · · · bim = ai (rbi1 · · · bij · · · bim )
i

i

showing the other containment. Proving the claim.
This is a contradiction to the fact that R satisfies τ -U-ACCP, proving we must in finitely
many steps arrive at a τ -U-α-factorization of a1 , proving R is indeed τ -U-α as desired. 
The following diagram summarizes our results from the Theorems 4.3 and 4.4 where ⋆
represents R being strongly associate, and † represents R is τ -U-refinable:
τ -α-β-UFR

ks
τ ❧-U-α-HFR
◗◗
19

⋆

τ -α-HFR

◗◗◗◗
❧❧❧
❧❧❧
◗◗◗†◗
❧
❧
❧
⋆
◗◗◗◗
❧❧❧
❧
❧
❧

$,
†
+3 τ -U-BFR
+3 τ -U-β-FFR
τ -U-α-β-UFR
KS
]e ❈❈❈
❈❈❈❈
❈❈❈❈
❈❈❈❈

❈❈
+3 τ -U-β-WFFR ❈❈❈❈❈❈❈
τ -β-WFFR
❈❈❈❈❈ τ -BFR
❈❈❈❈
❈❈❈❈
❈❈❈

+3 τ -U-α-β df ring
τ -α-β df ring
τ -β-FFR

†

+3

τ -U-ACCP
KS
τ −ACCP

†

+3

τ -U-α
KS
τ -α

We have left off the relations which were proven in [13, Theorem 4.1], and focused instead
on the rings satisfying the U-finite factorization properties. Examples given in [9, 10, 4, 2]
show that arrows can neither be reversed nor added to the diagram with a few exceptions.
Question 4.5. Does U-atomic imply atomic?
D.D. Anderson and S. Valdez-Leon show in [6, Theorem 3.13] that if R has a finite number
of non-associate irreducibles, then U-atomic and atomic are equivalent. This remains open

10

CHRISTOPHER PARK MOONEY

in general.
Question 4.6. Does U-ACCP imply ACCP?
We can modify M. Axtell’s proof of [9, Theorem 2.9] to add a partial converse to Theorem
4.4 (5) if τ is combinable and associate preserving. The idea is the same, but slight adjustments are required to adapt it to τ -factorizations and to allow uniqueness up to any type of
associate.
Theorem 4.7. Let β ∈ {associate, strongly associate, very strongly associate}. Let R be a
commutative ring with 1 and let τ be a symmetric relation on R# which is both combinable
and associate preserving. R is a τ -U-β-FFR if and only if R is a τ -U-β-WFFR.
Proof. (⇒) was already shown, so we need only prove the converse. (⇐) Suppose R is not
a τ -U-β-FFR. Let a ∈ R be a non-unit which has infinitely many τ -U-factorizations up to
β. Let b1 , b2 , . . . , bm be a complete list of essential τ -U-divisors of a up to β. Let
a = a1 · · · an ⌈c1 · · · ck ⌉ = a′1 · · · a′n′ ⌈d1 · · · dn ⌉
be two τ -U-factorizations of a and assume we have re-ordered the essential divisors in
both factorizations above so that the β of b1 appear first, followed by β of b2 , etc. Let
A = h(c1 ), (c2 ), . . . , (ck )i and B = h(d1 ), (d2 ), . . . , (dn )i be sequences of ideals. We call the
factorizations comparable if A is a subsequence of B or vice versa.
Suppose A is a proper subsequence of B
B = h(d1 ), . . . , (di1 ) = (c1 ), . . . , (di2 ) = (c2 ), . . . , (dik ) = (ck ), . . . , (dn )i
with n > k. Because τ is combinable and symmetric,
l
m
c
c
a = a′1 · · · a′n′ di1 di2 · · · dik (d1 · · · dc
d
·
·
·
d
·
·
·
d
)
i1 i2
ik
n

remains a τ -factorizations and [9, Lemma 1.3] ensures that this remains a U-factorization.
This yields
c
c
(a) = (d1 · · · dc
i1 di2 · · · dik · · · dn )(di1 di2 · · · dik ) = (d1 · · · dn ) = (c1 · · · ck )

= (c1 ) · · · (ck ) = (di1 ) · · · (dik ) = (di1 · · · dik ).
c
c
But then, (d1 · · · dc
i1 di2 · · · dik · · · dn ) cannot be an essential divisor, a contradiction, unless
n = k.
If n = k, then the sequences of ideals are identical, and we seek to prove this means the τ U-factorizations are the same up to β. It is certainly true for β = associate as demonstrated
in [9, Theorem 2.9]. So we have a pairing of the ci and di such that ci ∼ bj ∼ di for one of
the essential τ -U-divisors bj . We know further that ci and bj (resp. di and bj ) are β since R
is by assumption a τ -U-β-WFFR.
It is well established that β is transitive, so we can conclude that this same pairing
demonstrates that ci and di are β, not just associate. Thus the number of distinct τ -Ufactorizations up to β is less than or equal to the number of non-comparable finite sequences
of elements from the set {(b1 ), (b2 ), . . . , (bm )}.
From here we direct the reader to the proof of the second claim in [9, Theorem 2.9] where
it is shown that this set is finite.


GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

11

5. Direct Products
For each i, 1 ≤ i ≤ N, let Ri be commutative rings with τi a symmetric relation on Ri# .
We define a relation τ× on R = R1 × · · · × RN which preserves many of the theorems about
direct products from [8] for τ -factorizations. Let (ai ), (bi ) ∈ R# , then (ai )τ× (bi ) if and only
if whenever ai and bi are both non-units in Ri , then ai τi bi .
For convenience we will adopt the following notation: Suppose x ∈ Ri , then x(i) =
(1R1 , · · · , 1Ri−1 , x, 1Ri+1 , · · · 1RN ). so x appears in the ith coordinate, and all other entries
(1) (2)
(n)
are the identity. Thus for any (ai ) ∈ R, we have (ai ) = a1 a2 · · · an is a τ× -factorization.
We will always move any τ× -factors which may become units in this process to the front and
collect them there.
Lemma 5.1. Let R = R1 × · · · × RN for N ∈ N. Then (ai ) ∼ (bi ) (resp. (ai ) ≈ (bi )) if and
only if ai ∼ bi (resp. ai ≈ bi ) for every i. Furthermore, (ai ) ∼
= bi for all i,
= (bi ) implies ai ∼
∼
∼
and for ai , bi all non-zero, ai = bi for all i ⇒ (ai ) = (bi ).
Proof. See [6, Theorem 2.15].



Example 5.2. If ai0 = 0 for even one index 1 ≤ i0 ≤ N, then ai ∼
= bi for all i need not
imply (ai ) ∼
= (bi ).
Consider the ring R = Z × Z, with τi = Z# × Z# for i = 1, 2, the usual factorization. We
6= (0, 1).
have 1 ∼
= 0 since Z is a domain; however (0, 1) = (0, 1)(0, 1) shows (0, 1) ∼
= 1 and 0 ∼
Lemma 5.3. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# for
each i. Let α ∈ { irreducible, strongly irreducible, m-irreducible, very strongly irreducible}.
If (ai ) ∈ R is τ -α, then precisely one coordinate is not a unit.
Proof. Let a = (ai ) ∈ R be a non-unit which is τ× -α. Certainly not all coordinates can
be units, or else a ∈ U(R). Suppose for a moment there were at least two coordinates for
which ai is not a unit in Ri . After reordering, we may assume a1 and a2 are not units. Then
(1)
a = a1 (1R1 , a2 , · · · , aN ) is a τ× -factorization. But a is not even associate to either τ× -factor,
a contradiction.

Theorem 5.4. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# for
each i.
(1) A non-unit (ai ) ∈ R is τ× -atomic (resp. strongly atomic) if and only if ai0 is τi0 -atomic
(resp. strongly atomic) for some 1 ≤ i0 ≤ n and ai ∈ U(Ri ) for all i 6= i0 .
(2) A non-unit (ai ) ∈ R is τ× -m-atomic if and only if ai0 is τi0 -m-atomic for some 1 ≤ i0 ≤ n
and ai ∈ U(Ri ) for all i 6= i0 .
(3) A non-unit (ai ) ∈ R is τ× -very strongly atomic if and only if ai0 is τi0 -very strongly
atomic and non-zero for some 1 ≤ i0 ≤ n and ai ∈ U(Ri ) for all i 6= i0 .
Proof. (1) (⇒) Let a = (ai ) ∈ R be a non-unit which is τ× -atomic (resp. strongly atomic).
By Lemma 5.3, there is only one non-unit coordinate. Suppose after reordering if necessary
that a1 is the non-unit. If a1 were not τ1 -atomic (resp. strongly atomic), then there is a
τ1 -factorization, λ11 a11 a12 · · · a1k for which a1 6∼ a1j (resp. a1 6≈ a1j ) for any 1 ≤ j ≤ k. But
then
(1) (1)
(1)
(ai ) = (λ11 , a2 , . . . , an )a11 a12 · · · a1k
(1)

(1)

is a τ× -factorization. Furthermore, by Lemma 5.1 (ai ) 6∼ a1j (resp. (ai ) 6∼ a1j ) for all
1 ≤ j ≤ k. This would contradict the assumption that a was τ× -atomic (resp. strongly

12

CHRISTOPHER PARK MOONEY

atomic).
(⇐) Let a1 ∈ R1 , a non-unit with a1 being τ1 -atomic (resp. strongly atomic). Let
µi ∈ U(Ri ) for 2 ≤ i ≤ N. We show a = (a1 , µ2 , · · · µN ) is τ× -atomic (resp. strongly
atomic). Suppose a = (λ1 , . . . , λN )(a11 , . . . , a1N ) · · · (ak1 , . . . , akN ) is a τ× -factorization of a.
We first note aij ∈ U(Rj ) for all j ≥ 2. Furthermore, this means ai1 is not a unit in R1
for 1 ≤ i ≤ k, otherwise we would have units as factors in a τ× factorization. This means
a1 = λ1 a11 · · · ak1 is a τ1 factorization of a τ1 -atomic (resp. strongly atomic) element. Thus,
we must have a1 ∼ aj1 (resp. a1 ≈ aj1 ) for some 1 ≤ j ≤ k. Hence by Lemma 5.1, we have
a ∼ (aj1 , . . . , ajN ) (resp. a ≈ (aj1 , . . . , ajN ) for some 1 ≤ j ≤ k and a is τ× atomic (resp.
strongly atomic) as desired.
(2) (⇒) Let a = (ai ) ∈ R be a non-unit which is τ× -m-atomic. By Lemma 5.3, there is
only one non-unit coordinate, say a1 after reordering if necessary. Let a1 = λ11 a11 a12 · · · a1k
be a τ1 factorization for which a1 6∼ a1j0 for at least one 1 ≤ j0 ≤ k. But then
(1) (1)

(1)

(ai ) = (λ11 , a2 , . . . , an )a11 a12 · · · a1k

(1)

is a τ× -factorization of a for which (by Lemma 5.1) a = (ai ) 6∼ a1j . This contradicts the
0
hypothesis that a is τ× -m-atomic.
(⇐) Let a1 ∈ R1 , a non-unit with a1 being τ1 -m-atomic. Let µi ∈ U(Ri ) for 2 ≤ i ≤ N.
We show a = (a1 , µ2 , · · · µN ) is τ× -m-atomic. Suppose
a = (λ1 , . . . , λN )(a11 , . . . , a1N ) · · · (ak1 , . . . , akN )
is a τ× -factorization of a. We first note aij ∈ U(Rj ) for all j ≥ 2. As before, this means
a1 = λ1 a11 · · · ak1 is a τ1 factorization of a τ1 -m-atomic element. Hence a1 ∼ aj1 for each
1 ≤ j ≤ k. By Lemma 5.1 we have a ∼ (aj1 , . . . , ajN ) for all 1 ≤ j ≤ k and thus a is
τ× -m-atomic as desired.
(3) (⇒) Let a = (a1 , . . . aN ) be a non-unit which is τ× -very strongly atomic. By Lemma
5.3, we may assume a1 is the non-unit, and aj is a unit for j ≥ 2. We suppose for a moment
that a1 = 01 . But then (0, a2 , . . . aN ) = (0, 1, . . . 1) · (0, a2 , . . . aN ) shows that a ∼
6= a, a
∼
∼
contradiction. Lemma 5.1 shows that if a = a, then ai = ai for each 1 ≤ i ≤ N. Hence, if a1
were not τ1 -very strongly atomic, then there is a τ1 -factorization, λ11 a11 a12 · · · a1k for which
a1 ∼
6= a1j for any 1 ≤ j ≤ k. But then
(1) (1)

(1)

(ai ) = (λ11 , a2 , . . . , an )a11 a12 · · · a1k

is a τ× -factorization. Furthermore, since every coordinate is non-zero, by Lemma 5.1 (ai ) ∼
6=
(1)
a1j for all 1 ≤ j ≤ k. This would contradict the assumption that a was τ× -very strongly
atomic.
(⇐) Let a1 ∈ R1# be τ1 -very strongly atomic. Let µi ∈ U(Ri ) for 2 ≤ i ≤ N. We
show a = (a1 , µ2 , · · · µN ) is τ× -very strongly atomic. We first check a ∼
= a. By definition
of τ1 -very strongly atomic, a1 ∼
= a1 . Certainly as units, we have µi ∼
= µi for each i ≥ 2.
Lastly, all of these are non-zero, so we may apply Lemma 5.1 to see that a ∼
= a. Suppose
a = (λ1 , . . . , λN )(a11 , . . . , a1N ) · · · (ak1 , . . . , akN ) is a τ× -factorization of a. We first note
aij ∈ U(Rj ) for all j ≥ 2. As before, this means a1 = λ1 a11 · · · ak1 is a τ1 factorization of
a τ1 -very strongly atomic element. Hence a1 ∼
= aj1 for some 1 ≤ j ≤ k. By Lemma 5.1 we
have a ∼
(a
,
.
.
.
,
a
)
and
thus
a
is
τ
-very
strongly
atomic as desired.

= j1
jN
×
Lemma 5.5. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# . Let
α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible }. Then we

GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

13

have the following.
(1) If a = λa1 · · · aln ⌈b1 · · · bmm⌉ is a τi -U-α-factorization of some non-unit a ∈ Ri , then
(i)
(i)
(i)
(i)
a(i) = λ(i) a1 · · · an b1 · · · bm is a τ× -U-α-factorization.
(2) Conversely, let ai0 ∈ Ri0 be a non-unit and µi ∈ U(Ri ) for all i 6= i0 . Let
(µ1 , µ2 , . . . , µi0 −1 , ai0 , µi0 +1 . . . , µN ) = (λi )(a1i )(a2i ) · · · (ani ) ⌈(b1i )(b2i ) · · · (bmi )⌉
be a τ× -U-α-factorization. Then

is a τi0 -U-α-factorization.



ai0 = λi0 a1i0 · · · ani0 b1i0 · · · bi0

Proof. (1) Let a = λa1 · · · an ⌈b1 · · · bm ⌉lbe a τi -U-α-factorization
of some non-unit a ∈ Ri . It
m
(i)
(i)
(i)
(i)
(i) (i)
is easy to see that a = λ a1 · · · an b1 · · · bm is a τ× -factorization. Furthermore, bj 6= 0
(i)

for all 1 ≤ j ≤ or else it would not be a τi -factorization. Hence by Theorem 5.4 bj is τ× -α
for each 1 ≤ j ≤ m. Thus it suffices to show that we actually have a U-factorization.
Since a = λa1 · · · an ⌈b1 · · · bm ⌉ is a U-factorization, we know ak (b1 · · · bm ) = (b1 · · · bm ) for
all 1 ≤ k ≤ n. In the other coordinates, we have (1Rj ) = (1Rj ) for all j 6= i. Hence, we apply
(i) (i)
(i)
(i)
(i)
Lemma 5.1 and see that this implies that ak (b1 · · · bm ) = (b1 · · · bm ) for all 1 ≤ k ≤ n.
c
(i) (i)
(i)
(i)
Similarly we have bj (b1 · · · bbj · · · bm ) 6= (b1 · · · bbj · · · bm ) which implies bj (b1 · · · bj · · · bm ) 6=
c
(i)
(i)
(i)
(b1 · · · bj · · · bm ), so this is indeed a U-factorization.
(2) Let
(µ1 , µ2 , . . . , µi0 −1 , ai0 , µi0 +1 . . . , µN ) = (λi )(a1i )(a2i ) · · · (ani ) ⌈(b1i )(b2i ) · · · (bmi )⌉

be a τ× -U-α-factorization. We note that aji ∈ U(Ri ) for all i 6= i0 and all 1 ≤ j ≤ n and
bji ∈ U(Ri ) for all i 6= i0 and all 1 ≤ j ≤ m since they divide the unit µi . Next, every
coordinate in the i0 place must be a non-unit in Ri0 or else this factor would be a unit in R
and therefore could not occur as a factor in a τ× -factorization. This tells us that


ai0 = λi0 a1i0 · · · ani0 b1i0 · · · bi0

is a τi0 -factorization. Furthermore, (bki ) is assumed to be τ× -α for all 1 ≤ k ≤ m, and the
other coordinates are units, so bki0 is τi0 -α for all 1 ≤ k ≤ m by Theorem 5.4. Again, we
need only show that


ai0 = λi0 a1i0 a2i0 · · · ani0 b1i0 b2i0 · · · bmi0

is a U-factorization. Since all the coordinates other than i0 are units, we simply apply Lemma
5.1 and see that we indeed maintain a U-factorization.

Theorem 5.6. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# . Let
α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible}. Then R is
τ× -U-α if and only if Ri is τi -U-α for each 1 ≤ i ≤ N.

Proof. (⇒) Let a ∈ Ri0 be a non-unit. Then a(i0 ) is a non-unit in R and therefore has a τ× U-α-factorization. Furthermore, the only possible non-unit factors in this factorization must
occur in the ith
0 coordinate. Thus as in Lemma 5.5 (2), we have found a τi0 -U-α-factorization
of a by taking the product of the ith
0 entries. This shows Ri0 is τi0 -U-α as desired.

14

CHRISTOPHER PARK MOONEY

(⇐) Let a = (ai ) ∈ R be a non-unit. For each non-unit ai ∈ Ri , there is a τi -U-αfactorization of ai , say


ai = λi ai1 · · · aini bi1 · · · bimi .
(i)

If ai ∈ U(Ri ), then ai ∈ U(R) and we can simply collect these unit factors in the front, so
we need not worry about these factors. This yields a τ× -U-α-factorization
&m
'
n
Y
Y
(i) (i)
(i)
(i)
(i)
a = (ai ) =
λi ai1 · · · ain
bi1 · · · bim .
i

i=1

i

i=0

It is certainly a τ× -factorization. Furthermore, bjk 6= 0j for 1 ≤ j ≤ m and 1 ≤ k ≤ mj , so
(j)
bjk is τ× -α by Theorem 5.4. It is also clear from Lemma 5.5 that this is a U-factorization,
showing every non-unit in R has a τ× -U-α-factorization.


Theorem 5.7. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# .
Let α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible} and let
β ∈ {associate, strongly associate, very strongly associate}. Then R is a τ× -U-α-β-df ring if
and only if Ri is τi -U-α-β-df ring for each 1 ≤ i ≤ N.
Proof. (⇒) Let a ∈ Ri0 be a non-unit. Suppose there were an infinite number of τi0 -U-α
(i0 ) ∞
essential divisors of a, say {bj }∞
j=1 none of which are β. But then {bj }j=1 yields an infinite
set of τ× -U-α-divisors of a(i0 ) by Lemma 5.5. Furthermore, none of them are β by Lemma
5.1.
(⇐) Let (ai ) ∈ R be a non-unit. We look at the collection of τ× -U-α essential divisors of
(ai ). Each must be of the form (λ1 , · · · , bi0 , · · · λN ) with λi ∈ U(Ri ) for each i and with bi0
τi0 -α for some 1 ≤ i0 ≤ N. But then bi0 is a τi0 -α essential divisor of ai0 . For each i between
1 and N, Ri is a τi -U-α-β-df ring, so there can be only finitely many τi -α essential divisors
of ai up to β, say N(ai ). If ai ∈ Ri , then we can simply set N(ai ) = 0 since it is a unit and
has no non-trivial τi -U-factorizations. Hence there can be only
N((ai )) := N(a1 ) + N(a2 ) + · · · + N(aN ) =

N
X

N(ai )

i=1

τ× -α essential divisors of (ai ) up to β. This proves the claim.



Corollary 5.8. Let α and β be as in the theorem. Let R = R1 × · · · × RN for N ∈ N with
τi a symmetric relation on Ri# . Then R is a τ× -U-α τ× -U-α-β-df ring if and only if Ri is
τi -U-α τi -U-α-β-df ring for each 1 ≤ i ≤ N.
Proof. This is immediate from Theorem 5.7 and Theorem 5.6.



Theorem 5.9. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# . Then
R is a τ× -U-BFR if and only if Ri is a τi -U-BFR for every i.
Proof. (⇒) Let a ∈ Ri0 be a non-unit. Then a(i0 ) is a non-unit in R, and hence has a bound
on the number of essential divisors in any τ× -U-factorization, say Ne (a(i0 ) ). We claim this
also bounds the number of essential divisors in any τi0 -U-factorization of a. Suppose for a
moment a = a1 · · · an ⌈b1 · · · bm ⌉ were a τi0 -U-factorization with m > Ne (a(i) ). But then
l
m
(i0 )
(i0 ) (i0 )
(i0 )
(i0 )
a = λ a1 · · · an b1 · · · bm

GENERALIZED U-FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

15

is a τ× -U-factorization with more essential divisors than is allowed, a contradiction.
(⇐) Let a = (ai ) ∈ R be a non-unit. Let B(a) = max{Ne (ai )}N
i=1 . Where Ne (ai ) is
the number of essential divisors in any τi -U-factorization of ai , and will say for ai ∈ U(Ri ),
Ne (ai ) = 0. We claim that B(a)N is a bound on the number of essential divisors in any
τ× -U-factorization of a. Let
(ai ) = (λi )(a1i ) · · · (ani ) ⌈(b1i ) · · · (bmi )⌉
be a τ× -U-factorization. We can decompose this factorization so that each factor has at most
one non-unit entry as follows:
(ai ) =

N
Y
i=1

(i) (i)

λi a1i · · ·

N
Y
i=1

a(i)
ni

N
Y
i=1

(i)

b1i · · ·

N
Y

b(i)
mi .

i=1

Some of these factors may indeed be units; however, by allowing a unit factor in the front of
every τ -U-factorization, we simply combine all the units into one at the front, and maintain
a τ× -factorization. We can always rearrange this to be a τ× -U-factorization. Furthermore,
(i)
since aji is inessential, by Lemma 5.1 aji is inessential. Only some of the components of the
essential divisors could become inessential, for instance if one coordinate were a unit. At
(i)
worst when we decompose, bji remains an essential divisor for all 1 ≤ j ≤ m and for all
1 ≤ i ≤ N. But then the product of each of the ith coordinates gives a τi -U-factorization of
ai and thus is bounded by Ne (ai ), so we have m ≤ Ne (ai ) ≤ B(a) and therefore there are no
more than B(a)N essential divisors. Certainly the original factorization is no longer than
the one we constructed through the decomposition, proving the claim and completing the
proof.

Theorem 5.10. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# .
Let α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible }. Then R
is τ× -U-α-HFR if and only if Ri is a τi -U-α-HFR for each i.
Proof. (⇒) Let a ∈ Ri0 be a non-unit. We know by Theorem 5.6 Then a(i0 ) is a non-unit
in R, and has an τ× -U-α-factorization. Suppose there were τi0 -U-α-factorizations of a with
different numbers of essential divisors, say:
a = λa1 · · · an ⌈b1 · · · bm ⌉ = µc1 · · · cn′ ⌈d1 · · · dm′ ⌉
where m 6= m′ . By Lemma 5.5 this yields two τ× -U-α-factorizations:
l
m
l
m
(i0 )
(i0 )
(i0 )
(i0 )
(i0 )
(i0 ) (i0 )
(i0 )
(i0 )
(i0 ) (i0 )
a = λ a1 · · · an b1 · · · bn
= µ c1 · · · cn ′ d 1 · · · d n ′ .

This contradicts the hypothesis that R is a R is τ× -U-α-HFR.
(⇐) Let (ai ) ∈ R be a non-unit. Suppose we had two τ× -U-α factorizations
m
l
′
′
′
′
′
′
(ai ) = (λi )(a1i )(a2i ) · · · (ani ) ⌈(b1i )(b2i ) · · · (bmi )⌉ = (µi )(a1i )(a2i ) · · · (an′i ) (b1i )(b2i ) · · · (bm′i ) .

For each i0 , if ai0 is a non-unit in Ri0 , then since each τ× -α element can only have one
coordinate which is not a unit, we can simply collect all the τ× -divisors which have the i0
coordinate a non-unit. This product forms a τi0 -U-α-factorization of ai0 and therefore the
number of essential τ× -factors with coordinate i0 a non-unit must be the same in the two
factorizations. This is true for each coordinate i0 , hence m = m′ as desired.


16

CHRISTOPHER PARK MOONEY

Theorem 5.11. Let R = R1 × · · · × RN for N ∈ N with τi a symmetric relation on Ri# .
Let α ∈ {irreducible, strongly irreducible, m-irreducible, very strongly irreducible} and let
β ∈ {associate, strongly associate }. Then R is τ× -U-α-β-UFR if and only if Ri is a τi -Uα-β-UFR for each i.
Proof. We simply apply Lemma 5.1 to the proof of Theorem 5.10, to see that the factors can
always be rearranged to match associates of the correct type.

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Reinhart Center, Viterbo University, 900 Viterbo Drive, La Crosse, WI 54601
E-mail address: cpmooney@viterbo.edu

