text: τ-COMPLETE FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS 13 ≥ for each i. We note here that n 1 or else we would have (a ) = (a ). Because τ is i i i+1 reﬁnable, we can create τ-factorizations as follows: · · · · · · · · · · · · a = λ a b b = λ λ a b b b b = . 1 1 2 11 1n1 1 2 3 21 2n2 11† 1n1 After N(a ) iterations, we will arrive at a τ-factorization, , of length at least N(a ) since 1 1 at each stage the length increases by at least 1. Now τ is reﬁnable and R is τ-complete, so † we apply Theorem 3.4 to τ-reﬁne the τ-factorization, , of length N(a ) into a τ-complete 1 † factorization. (resp. Because R is τ-completable, we can τ-reﬁne the factorization, , into a τ-complete factorization.) This can only increase the length of the factorization which contradicts the fact that N(a ) is the bound on the length of τ-complete factorizations of 1 (cid:3) a . 1 We get a similar analogue for τ-atomicable, strongly atomicable, m-atomicable, unreﬁn- ably atomicable, very strongly atomicable rings. Theorem 4.5. Let R be a commutative ring with 1 and τ be a symmetric relation on R#. ∈ { Let α atomicable, strongly atomicable, m-atomicable, unreﬁnably atomicable, very strongly } atomicable . If α = atomicable (resp. strongly atomicable, m-atomicable, unreﬁnably atom- (cid:48) icable, very strongly atomicable), set α = atomic (resp. strongly atomic, m-atomic, unre- ∈ { ﬁnably atomic, very strongly atomic). Let β associate, strongly associate, very strongly } associate . Then we have the following. (1) If R is a τ-α-β-UFR, then R is a τ-α-HFR. (2) If τ is reﬁnable and R is a τ-α-β-UFR, then R is a τ-β-FFR. (cid:48) (3) If R is a τ-α-β-UFR, then R is a τ-α τ-α -β-divisor ﬁnite ring. (4) If τ is reﬁnable and R is a τ-α-β-HFR, then R is a τ-BFR. (cid:48) (cid:48) (5) If R is a τ-α τ-α -β-divisor ﬁnite ring, then R is a τ-α -β-divisor ﬁnite ring. (cid:48) (6) If τ is reﬁnable and R is a τ-β-WFFR, then R is a τ-α τ-α -β-divisor ﬁnite ring. The following diagram summarizes the relationship between these new ﬁnite factorization ∇ properties that R might possess, where indicates τ is reﬁnable: τ(cid:48)(cid:56) -α-HFR ∇ (cid:36)(cid:44) ∇ ∇ (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) τ-α-β-UFR τ-complete-β-FFR τ-BFR τ-ACCP (cid:11)(cid:19) (cid:11)(cid:19) τ-β-WFFR τ-α ∇ (cid:11)(cid:19) (cid:112)(cid:120) (cid:11)(cid:19) (cid:48) (cid:43)(cid:51) (cid:48) τ-α τ-α -β-df ring τ-α -β-df ring Proof. (1) Suppose R is a τ-α-β-UFR. By hypothesis R is τ-α. Let a be a non-unit. Suppose · · · · · · (cid:48) a = λa a = µb b were two τ-α factorizations with diﬀerent lengths. Then this 1 n 1 m contradicts the fact that R is a τ-α-β-UFR, and proves the theorem. (cid:48) (2) Suppose τ is reﬁnable and R is a τ-α-β-UFR. Then R is a τ-α -β-UFR by Corollary 4.2. We then apply Theorem 2.4 and the hypothesis that τ is reﬁnable to conclude that R is also a τ-β-FFR. (cid:48) (3) Let R be a τ-α-β-UFR. Then R is τ-α and again, R is a τ-α -β-UFR by Corollary 4.2. ∈ · · · (cid:48) Let a R be a non-unit. Let a = λa a be the unique τ-α -factorization up to β. Then 1 n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: τ-COMPLETE

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: for

text: each

text: i.

text: We

text: note

text: here

text: that

text: n

text: or

text: else

text: we

text: would

text: have

text: (a

text: (a

text: Because

text: τ

text: is

text: i

text: i

text: reﬁnable,

text: we

text: can

text: create

text: τ-factorizations

text: as

text: follows:

text: a

text: λ

text: a

text: b

text: b

text: λ

text: λ

text: a

text: b

text: b

text: b

text: b

text: After

text: N(a

text: iterations,

text: we

text: will

text: arrive

text: at

text: a

text: τ-factorization,

text: of

text: length

text: at

text: least

text: N(a

text: since

text: at

text: each

text: stage

text: the

text: length

text: increases

text: by

text: at

text: least

text: Now

text: τ

text: is

text: reﬁnable

text: and

text: R

text: is

text: τ-complete,

text: so

text: we

text: apply

text: Theorem

text: to

text: τ-reﬁne

text: the

text: τ-factorization,

text: of

text: length

text: N(a

text: into

text: a

text: τ-complete

text: factorization.

text: (resp.

text: Because

text: R

text: is

text: τ-completable,

text: we

text: can

text: τ-reﬁne

text: the

text: factorization,

text: into

text: a

text: τ-complete

text: factorization.)

text: This

text: can

text: only

text: increase

text: the

text: length

text: of

text: the

text: factorization

text: which

text: contradicts

text: the

text: fact

text: that

text: N(a

text: is

text: the

text: bound

text: on

text: the

text: length

text: of

text: τ-complete

text: factorizations

text: of

text: a

text: We

text: get

text: a

text: similar

text: analogue

text: for

text: τ-atomicable,

text: strongly

text: atomicable,

text: m-atomicable,

text: unreﬁn-

text: ably

text: atomicable,

text: very

text: strongly

text: atomicable

text: rings.

text: Theorem

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: τ

text: be

text: a

text: symmetric

text: relation

text: on

text: R#.

text: Let

text: α

text: atomicable,

text: strongly

text: atomicable,

text: m-atomicable,

text: unreﬁnably

text: atomicable,

text: very

text: strongly

text: atomicable

text: If

text: α

text: atomicable

text: (resp.

text: strongly

text: atomicable,

text: m-atomicable,

text: unreﬁnably

text: atom-

text: icable,

text: very

text: strongly

text: atomicable),

text: set

text: α

text: atomic

text: (resp.

text: strongly

text: atomic,

text: m-atomic,

text: unre-

text: ﬁnably

text: atomic,

text: very

text: strongly

text: atomic).

text: Let

text: β

text: associate,

text: strongly

text: associate,

text: very

text: strongly

text: associate

text: Then

text: we

text: have

text: the

text: following.

text: If

text: R

text: is

text: a

text: τ-α-β-UFR,

text: then

text: R

text: is

text: a

text: τ-α-HFR.

text: If

text: τ

text: is

text: reﬁnable

text: and

text: R

text: is

text: a

text: τ-α-β-UFR,

text: then

text: R

text: is

text: a

text: τ-β-FFR.

text: If

text: R

text: is

text: a

text: τ-α-β-UFR,

text: then

text: R

text: is

text: a

text: τ-α

text: τ-α

text: -β-divisor

text: ﬁnite

text: ring.

text: If

text: τ

text: is

text: reﬁnable

text: and

text: R

text: is

text: a

text: τ-α-β-HFR,

text: then

text: R

text: is

text: a

text: τ-BFR.

text: If

text: R

text: is

text: a

text: τ-α

text: τ-α

text: -β-divisor

text: ﬁnite

text: ring,

text: then

text: R

text: is

text: a

text: τ-α

text: -β-divisor

text: ﬁnite

text: ring.

text: If

text: τ

text: is

text: reﬁnable

text: and

text: R

text: is

text: a

text: τ-β-WFFR,

text: then

text: R

text: is

text: a

text: τ-α

text: τ-α

text: -β-divisor

text: ﬁnite

text: ring.

text: The

text: following

text: diagram

text: summarizes

text: the

text: relationship

text: between

text: these

text: new

text: ﬁnite

text: factorization

text: properties

text: that

text: R

text: might

text: possess,

text: where

text: indicates

text: τ

text: is

text: reﬁnable:

text: -α-HFR

text: τ-α-β-UFR

text: τ-complete-β-FFR

text: τ-BFR

text: τ-ACCP

text: τ-β-WFFR

text: τ-α

text: τ-α

text: τ-α

text: -β-df

text: ring

text: τ-α

text: -β-df

text: ring

text: Proof.

text: Suppose

text: R

text: is

text: a

text: τ-α-β-UFR.

text: By

text: hypothesis

text: R

text: is

text: τ-α.

text: Let

text: a

text: be

text: a

text: non-unit.

text: Suppose

text: a

text: λa

text: a

text: µb

text: b

text: were

text: two

text: τ-α

text: factorizations

text: with

text: diﬀerent

text: lengths.

text: Then

text: this

text: n

text: m

text: contradicts

text: the

text: fact

text: that

text: R

text: is

text: a

text: τ-α-β-UFR,

text: and

text: proves

text: the

text: theorem.

text: Suppose

text: τ

text: is

text: reﬁnable

text: and

text: R

text: is

text: a

text: τ-α-β-UFR.

text: Then

text: R

text: is

text: a

text: τ-α

text: -β-UFR

text: by

text: Corollary

text: We

text: then

text: apply

text: Theorem

text: and

text: the

text: hypothesis

text: that

text: τ

text: is

text: reﬁnable

text: to

text: conclude

text: that

text: R

text: is

text: also

text: a

text: τ-β-FFR.

text: Let

text: R

text: be

text: a

text: τ-α-β-UFR.

text: Then

text: R

text: is

text: τ-α

text: and

text: again,

text: R

text: is

text: a

text: τ-α

text: -β-UFR

text: by

text: Corollary

text: Let

text: a

text: R

text: be

text: a

text: non-unit.

text: Let

text: a

text: λa

text: a

text: be

text: the

text: unique

text: τ-α

text: -factorization

text: up

text: to

text: β.

text: Then

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: τ-COMPLETE FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS 15 ﬁnite number of ways of grouping the factors to generate diﬀerent τ-factorizations up to β. · · · Given a τ-factorization a = λa a , there is a τ-complete factorization for each a with ≤ ≤ · ·1· n ≤ ≤ i 1 i n. Suppose a = λ b im is the τ-complete factorization of a for 1 i n. By i i i1 i i hypothesis, τ is reﬁnable, so · · · · · · · · · · · · · · · · (3) a = (λλ λ )b b b b b b 1 n 11 1m1 21 2m2 n1 nmn is a τ-factorization. By Theorem 3.4, this is a τ-complete factorization and hence was among the ﬁnite number of τ-complete factorizations of a up to β. (4) This proof is nearly identical to the proof of (3). The only modiﬁcation is that we can use the fact that since R is τ-completable to automatically conclude that any factorization · · · (cid:3) a = λa a can be τ-reﬁned into a τ-complete factorization of the form of Equation 3. 1 n Theorem 4.8. Let R be a commutative ring with 1 and τ be a symmetric relation on R#. ∈ { } Let β associate, strongly associate, very strongly associate . We have the following. (1) If R is a β-WFFR, then R is a τ-β-WFFR. (2) If R is a τ-β-WFFR, then R is a τ-complete-β-divisor ﬁnite ring. (3) If R is a τ-β-atomic divisor ﬁnite ring, then R is a τ-complete-β-divisor ﬁnite ring. (4) If R is a τ-β-strongly atomic divisor ﬁnite ring, then R is a τ-complete-β-divisor ﬁnite ring. (5) If τ is reﬁnable and R is a τ-β-m-atomic divisor ﬁnite ring, then R is a τ-complete-β- divisor ﬁnite ring. ∈ Proof. (1) Let a R be a non-unit. If a has a ﬁnite number of divisors up to β, then it certainly has a ﬁnite number τ-divisors up to β. ∈ (2) Let a R be a non-unit. If there are a ﬁnite number of τ-divisors of a up to β, then certainly there are a ﬁnite number of τ-divisors which occur as a τ-factor in some τ- complete-factorization of a up to β. ∈ { }∞ (3) (resp. (4)) Let a R be a non-unit. Suppose a is a inﬁnite collection of non-β τ- i i=1 · · · divisors which occur in some τ-complete factorization of a. Say a = λ a b b is one such i i i1 ini τ-complete factorization. By Theorem 3.1, this τ-complete factorization is τ-atomic (resp. strongly atomic). This provides an inﬁnite number of non-β τ-atomic (resp. τ-strongly atomic) divisors of a, a contradiction. ∈ { }∞ (5) Let a R be a non-unit. Suppose a is a inﬁnite collection of non-β τ-divisors i i=1 · · · which occur in some τ-complete factorization of a. Say a = λ a b b is one such τ- i i i1 ini complete factorization. We have a τ which is reﬁnable, so by Theorem 3.1, this τ-complete factorization is τ-m-atomic. This provides an inﬁnite number of non-β τ-m-atomic divisors (cid:3) of a, a contradiction. We notice at this point that many of the τ-ﬁnite factorization and τ-complete ﬁnite fac- ∈ torization properties result in R having the property that for a given non-unit a R, there is a ﬁnite number of divisors of a which occur as a τ-factor of some τ-complete factorization. We summarize these in the form of the following corollary. Corollary 4.9. Let R be a commutative ring with 1 and τ be a symmetric relation on R#. ∈ { } Let β associate, strongly associate, very strongly associate . If R satisﬁes any of the following conditions, then R is a τ-complete-β-divisor ﬁnite ring. (1) R is a β-FFR.

text: τ-COMPLETE

label: paragraph

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: ﬁnite

text: number

text: of

text: ways

text: of

text: grouping

text: the

text: factors

text: to

text: generate

text: diﬀerent

text: τ-factorizations

text: up

text: to

text: β.

text: Given

text: a

text: τ-factorization

text: a

text: λa

text: a

text: there

text: is

text: a

text: τ-complete

text: factorization

text: for

text: each

text: a

text: with

text: ·1·

text: n

text: i

text: i

text: n.

text: Suppose

text: a

text: λ

text: b

text: im

text: is

text: the

text: τ-complete

text: factorization

text: of

text: a

text: for

text: i

text: n.

text: By

text: i

text: i

text: i

text: i

text: hypothesis,

text: τ

text: is

text: reﬁnable,

text: so

label: equation

text: a

text: (λλ

text: λ

text: )b

text: b

text: b

text: b

text: b

text: b

text: n

text: nmn

text: is

text: a

text: τ-factorization.

text: By

text: Theorem

text: this

text: is

text: a

text: τ-complete

text: factorization

text: and

text: hence

text: was

text: among

text: the

text: ﬁnite

text: number

text: of

text: τ-complete

text: factorizations

text: of

text: a

text: up

text: to

text: β.

text: This

text: proof

text: is

text: nearly

text: identical

text: to

text: the

text: proof

text: of

text: The

text: only

text: modiﬁcation

text: is

text: that

text: we

text: can

text: use

text: the

text: fact

text: that

text: since

text: R

text: is

text: τ-completable

text: to

text: automatically

text: conclude

text: that

text: any

text: factorization

text: a

text: λa

text: a

text: can

text: be

text: τ-reﬁned

text: into

text: a

text: τ-complete

text: factorization

text: of

text: the

text: form

text: of

text: Equation

text: n

text: Theorem

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: τ

text: be

text: a

text: symmetric

text: relation

text: on

text: R#.

text: Let

text: β

text: associate,

text: strongly

text: associate,

text: very

text: strongly

text: associate

text: We

text: have

text: the

text: following.

text: If

text: R

text: is

text: a

text: β-WFFR,

text: then

text: R

text: is

text: a

text: τ-β-WFFR.

text: If

text: R

text: is

text: a

text: τ-β-WFFR,

text: then

text: R

text: is

text: a

text: τ-complete-β-divisor

text: ﬁnite

text: ring.

text: If

text: R

text: is

text: a

text: τ-β-atomic

text: divisor

text: ﬁnite

text: ring,

text: then

text: R

text: is

text: a

text: τ-complete-β-divisor

text: ﬁnite

text: ring.

text: If

text: R

text: is

text: a

text: τ-β-strongly

text: atomic

text: divisor

text: ﬁnite

text: ring,

text: then

text: R

text: is

text: a

text: τ-complete-β-divisor

text: ﬁnite

text: ring.

text: If

text: τ

text: is

text: reﬁnable

text: and

text: R

text: is

text: a

text: τ-β-m-atomic

text: divisor

text: ﬁnite

text: ring,

text: then

text: R

text: is

text: a

text: τ-complete-β-

text: divisor

text: ﬁnite

text: ring.

text: Proof.

text: Let

text: a

text: R

text: be

text: a

text: non-unit.

text: If

text: a

text: has

text: a

text: ﬁnite

text: number

text: of

text: divisors

text: up

text: to

text: β,

text: then

text: it

text: certainly

text: has

text: a

text: ﬁnite

text: number

text: τ-divisors

text: up

text: to

text: β.

text: Let

text: a

text: R

text: be

text: a

text: non-unit.

text: If

text: there

text: are

text: a

text: ﬁnite

text: number

text: of

text: τ-divisors

text: of

text: a

text: up

text: to

text: β,

text: then

text: certainly

text: there

text: are

text: a

text: ﬁnite

text: number

text: of

text: τ-divisors

text: which

text: occur

text: as

text: a

text: τ-factor

text: in

text: some

text: τ-

text: complete-factorization

text: of

text: a

text: up

text: to

text: β.

text: (resp.

text: Let

text: a

text: R

text: be

text: a

text: non-unit.

text: Suppose

text: a

text: is

text: a

text: inﬁnite

text: collection

text: of

text: non-β

text: τ-

text: i

text: divisors

text: which

text: occur

text: in

text: some

text: τ-complete

text: factorization

text: of

text: a.

text: Say

text: a

text: λ

text: a

text: b

text: b

text: is

text: one

text: such

text: i

text: i

text: ini

text: τ-complete

text: factorization.

text: By

text: Theorem

text: this

text: τ-complete

text: factorization

text: is

text: τ-atomic

text: (resp.

text: strongly

text: atomic).

text: This

text: provides

text: an

text: inﬁnite

text: number

text: of

text: non-β

text: τ-atomic

text: (resp.

text: τ-strongly

text: atomic)

text: divisors

text: of

text: a,

text: a

text: contradiction.

text: Let

text: a

text: R

text: be

text: a

text: non-unit.

text: Suppose

text: a

text: is

text: a

text: inﬁnite

text: collection

text: of

text: non-β

text: τ-divisors

text: i

text: which

text: occur

text: in

text: some

text: τ-complete

text: factorization

text: of

text: a.

text: Say

text: a

text: λ

text: a

text: b

text: b

text: is

text: one

text: such

text: τ-

text: i

text: i

text: ini

text: complete

text: factorization.

text: We

text: have

text: a

text: τ

text: which

text: is

text: reﬁnable,

text: so

text: by

text: Theorem

text: this

text: τ-complete

text: factorization

text: is

text: τ-m-atomic.

text: This

text: provides

text: an

text: inﬁnite

text: number

text: of

text: non-β

text: τ-m-atomic

text: divisors

text: of

text: a,

text: a

text: contradiction.

text: We

text: notice

text: at

text: this

text: point

text: that

text: many

text: of

text: the

text: τ-ﬁnite

text: factorization

text: and

text: τ-complete

text: ﬁnite

text: fac-

text: torization

text: properties

text: result

text: in

text: R

text: having

text: the

text: property

text: that

text: for

text: a

text: given

text: non-unit

text: a

text: R,

text: there

text: is

text: a

text: ﬁnite

text: number

text: of

text: divisors

text: of

text: a

text: which

text: occur

text: as

text: a

text: τ-factor

text: of

text: some

text: τ-complete

text: factorization.

text: We

text: summarize

text: these

text: in

text: the

text: form

text: of

text: the

text: following

text: corollary.

text: Corollary

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: τ

text: be

text: a

text: symmetric

text: relation

text: on

text: R#.

text: Let

text: β

text: associate,

text: strongly

text: associate,

text: very

text: strongly

text: associate

text: If

text: R

text: satisﬁes

text: any

text: of

text: the

text: following

text: conditions,

text: then

text: R

text: is

text: a

text: τ-complete-β-divisor

text: ﬁnite

text: ring.

text: R

text: is

text: a

text: β-FFR.


text: 8 CHRISTOPHER PARK MOONEY Z { ± ± ± ± ± ± ± ± } (3) Let R = and let τ = ( 2, 2), ( 3, 3), ( 4, 9), ( 9, 4) . Then τ is associate preserving and symmetric, but not reﬁnable. We consider · the factorization 36 = 4 9 and notice that 4 and 9 are not even τ-irreducible since · · 4 = 2 2 and 9 = 3 3 are τ-factorizations which show 4 and 9 are not τ-atomic. On ± (cid:54) ± ± the other hand, we see that this factorization is τ-complete since 2 τ 3, 9 and ± (cid:54) ± ± 3 τ 2, 4, so no τ-reﬁnements of this factorization are valid τ-factorizations, so the factorization is τ-complete despite not being τ-atomic, strongly atomic, m-atomic, unreﬁnably atomic, or very strongly atomic. Examples given in [4] show that the other arrows are not reversible even when × τ = R# R#. As in a series of papers by A. Bouvier, [11, 12, 13, 14], a commutative ring is said to be ∈ ∈ pr´esimpliﬁable if x = xy for some x, y R implies x = 0 or y U(R). A nice property of the various τ-irreducibles deﬁned in [19] is that they all coincide when a ring is pr´esimpliﬁable. When R is pr´esimpliﬁable and τ is reﬁnable, we can add τ-complete factorizations to the list of equivalent τ-irreducible factorizations. We summarize this in the following theorem which follows relatively easily from Theorem 3.1. Theorem 3.3. Let R be pr´esimpliﬁable and τ be a symmetric, reﬁnable relation on R#. For · · · a τ-factorization a = λa a , the following are equivalent. 1 n (1) This is a τ-very strongly atomic factorization. (2) This is a τ-unreﬁnably atomic factorization. (3) This is a τ-complete factorization. (4) This is a τ-strongly atomic factorization. (5) This is a τ-m-atomic factorization. (6) This is a τ-atomic factorization. ∼ ∼ ⇒ Proof. In a pr´esimpliﬁable r∼ing, all the associate relations coincide. We have x y x = y, ∼ ⇒ ⇒ in particular a a a = a , so we have (5) (1). This coupled with what was shown i i i i (cid:3) in Theorem 3.1 completes the proof. Theorem 3.4. Let R be a commutative ring with 1 and let τ be a reﬁnable, symmetric · · · · · · relation on R#. If a = λa a is a τ-factorization and a = λ b b are τ-complete ≤ ≤ 1 n i i i1 imi factorizations for 1 i n, then · · · · · · · · · · · · · · · (1) a = (λλ λ )b b b b b b 1 n 11 1m1 21 2m2 n1 nmn is a τ-complete factorization. Proof. Because τ is reﬁnable, the factorization given in 1 is certainly a τ-factorization. It remains to be seen that this factorization is τ-complete. Taking the notation from the ≤ ≤ statement of the theorem, we suppose there is a τ-reﬁnement of b for some 1 i n and ≤ ≤ · · · ij 1 j m of the form b = µc c which makes the factorization in equation (1) properly i ij 1 k · · · longer. This also yields a τ-reﬁnement of the τ-complete factorization a = λ b b · · · · · · · · · i i i1 imi a = (λ µ)b b − c c b b i i i1 i(j 1) 1 k i(j+1) imi (cid:3) into a τ-factorization which is properly longer, a contradiction, completing the proof.

text: CHRISTOPHER

text: PARK

text: MOONEY

text: Z

label: list

text: Let

text: R

text: and

text: let

text: τ

text: Then

text: τ

text: is

text: associate

text: preserving

text: and

text: symmetric,

text: but

text: not

text: reﬁnable.

text: We

text: consider

text: the

text: factorization

text: and

text: notice

text: that

text: and

text: are

text: not

text: even

text: τ-irreducible

text: since

text: and

text: are

text: τ-factorizations

text: which

text: show

text: and

text: are

text: not

text: τ-atomic.

text: On

font: default

text: the

text: other

text: hand,

text: we

text: see

text: that

text: this

text: factorization

text: is

text: τ-complete

text: since

text: τ

text: and

text: τ

text: so

text: no

text: τ-reﬁnements

text: of

text: this

text: factorization

text: are

text: valid

text: τ-factorizations,

text: so

text: the

text: factorization

text: is

text: τ-complete

text: despite

text: not

text: being

text: τ-atomic,

text: strongly

text: atomic,

text: m-atomic,

text: unreﬁnably

text: atomic,

text: or

text: very

text: strongly

text: atomic.

text: Examples

text: given

text: in

text: show

text: that

text: the

text: other

text: arrows

text: are

text: not

text: reversible

text: even

text: when

text: τ

text: R#

text: R#.

text: As

text: in

text: a

text: series

text: of

text: papers

text: by

text: A.

text: Bouvier,

text: a

text: commutative

text: ring

text: is

text: said

text: to

text: be

text: pr´esimpliﬁable

text: if

text: x

text: xy

text: for

text: some

text: x,

text: y

text: R

text: implies

text: x

text: or

text: y

text: U(R).

text: A

text: nice

text: property

text: of

text: the

text: various

text: τ-irreducibles

text: deﬁned

text: in

text: is

text: that

text: they

text: all

text: coincide

text: when

text: a

text: ring

text: is

text: pr´esimpliﬁable.

text: When

text: R

text: is

text: pr´esimpliﬁable

text: and

text: τ

text: is

text: reﬁnable,

text: we

text: can

text: add

text: τ-complete

text: factorizations

text: to

text: the

text: list

text: of

text: equivalent

text: τ-irreducible

text: factorizations.

text: We

text: summarize

text: this

text: in

text: the

text: following

text: theorem

text: which

text: follows

text: relatively

text: easily

text: from

text: Theorem

text: Theorem

text: Let

text: R

text: be

text: pr´esimpliﬁable

text: and

text: τ

text: be

text: a

text: symmetric,

text: reﬁnable

text: relation

text: on

text: R#.

text: For

text: a

text: τ-factorization

text: a

text: λa

text: a

text: the

text: following

text: are

text: equivalent.

text: n

text: This

text: is

text: a

text: τ-very

text: strongly

text: atomic

text: factorization.

text: This

text: is

text: a

text: τ-unreﬁnably

text: atomic

text: factorization.

text: This

text: is

text: a

text: τ-complete

text: factorization.

text: This

text: is

text: a

text: τ-strongly

text: atomic

text: factorization.

text: This

text: is

text: a

text: τ-m-atomic

text: factorization.

text: This

text: is

text: a

text: τ-atomic

text: factorization.

text: Proof.

text: In

text: a

text: pr´esimpliﬁable

text: r∼ing,

text: all

text: the

text: associate

text: relations

text: coincide.

text: We

text: have

text: x

text: y

text: x

text: y,

text: in

text: particular

text: a

text: a

text: a

text: a

text: so

text: we

text: have

text: This

text: coupled

text: with

text: what

text: was

text: shown

text: i

text: i

text: i

text: i

text: in

text: Theorem

text: completes

text: the

text: proof.

text: Theorem

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: let

text: τ

text: be

text: a

text: reﬁnable,

text: symmetric

text: relation

text: on

text: R#.

text: If

text: a

text: λa

text: a

text: is

text: a

text: τ-factorization

text: and

text: a

text: λ

text: b

text: b

text: are

text: τ-complete

text: n

text: i

text: i

text: imi

text: factorizations

text: for

text: i

text: n,

text: then

text: a

text: (λλ

text: λ

text: )b

text: b

text: b

text: b

text: b

text: b

text: n

text: nmn

text: is

text: a

text: τ-complete

text: factorization.

text: Proof.

text: Because

text: τ

text: is

text: reﬁnable,

text: the

text: factorization

text: given

text: in

text: is

text: certainly

text: a

text: τ-factorization.

text: It

text: remains

text: to

text: be

text: seen

text: that

text: this

text: factorization

text: is

text: τ-complete.

text: Taking

text: the

text: notation

text: from

text: the

text: statement

text: of

text: the

text: theorem,

text: we

text: suppose

text: there

text: is

text: a

text: τ-reﬁnement

text: of

text: b

text: for

text: some

text: i

text: n

text: and

text: ij

text: j

text: m

text: of

text: the

text: form

text: b

text: µc

text: c

text: which

text: makes

text: the

text: factorization

text: in

text: equation

text: properly

text: i

text: ij

text: k

text: longer.

text: This

text: also

text: yields

text: a

text: τ-reﬁnement

text: of

text: the

text: τ-complete

text: factorization

text: a

text: λ

text: b

text: b

text: i

text: i

text: imi

text: a

text: (λ

text: µ)b

text: b

text: c

text: c

text: b

text: b

text: i

text: i

text: i(j

text: k

text: imi

text: into

text: a

text: τ-factorization

text: which

text: is

text: properly

text: longer,

text: a

text: contradiction,

text: completing

text: the

text: proof.


text: τ-REGULAR FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS 15 (2) Let R be a r-FFR, but suppose that R is not a τ-r-FFR. We then have a regular ∈ element a Reg(R)# that has an inﬁnite number of τ-r-factorizations up to rearrangement and associate, but again each of these is also r-factorization and are still unique up to rearrangement and associates which contradicts the fact that R is a r-FFR. ∈ (3) Let a Reg(R). Every τ-r-divisor divisor is a regular divisor of a, so there can be only ﬁnitely many up to associate. { }∞ ∈ | (4) Suppose we have an inﬁnite sequence a , a Reg(R)# for all k with a a but (cid:54)∼ ≥ i i=1 | k ∈ n+1 τ n(cid:54)∼ a a for all n 1. But then we still have a a , a Reg(R)# for all k but a a n+1 n n+1 τ n k n+1 (cid:3)n so we contradict r-ACCP. Concluding the proof. Corollary 5.3. The r-UFRs, r-FFRs, r-HFRs, r-BFRs as deﬁned in [6, Section 5] satisfy r- ACCP, and therefore τ-r-ACCP. Hence for τ reﬁnable, each is τ-r-atomic by Theorem 5.2 and Theorem 3.3. ∇ The following diagram summarizes our results ( represents τ being reﬁnable): τ(cid:50)(cid:58)-r-HFR r-BFR r-ACCP ∇ (cid:36)(cid:44) (cid:11)(cid:19) (cid:11)(cid:19) ∇ ∇ ∇ (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) τ-r-UFR τ(cid:50)(cid:58)-r-F(cid:75)(cid:83) FR τ-r-BFR τ-r-A(cid:75)(cid:83) CCP τ-r-atomic ∇ (cid:11)(cid:19) r-FFR τ-r(cid:50)(cid:58) -W(cid:75)(cid:83) FFR τ-ACCP ∇ (cid:11)(cid:19) (cid:36)(cid:44) (cid:43)(cid:51) r-WFFR τ-atomic τ-r-idf τ-r-idf Lemma 5.4. Let R be a commutative ring with 1 and let τ be a symmetric relation on R#. Let ∈ {∅ } α , atomic, strongly atomic, m-atomic, unreﬁnably atomic, very strongly atomic . Every non-unit element in a τ -U-α-factorization is an essential divisor. Moreover, given a τ -α- reg ∅ reg factorization, every τ-factor is essential. When α = , we mean simply a τ -U-factorization. reg ∈ · · · (cid:100) · · · (cid:101) Proof. Let a R be a non-unit and let a = λa a b b be a τ -U-α-factorization. · · · · · · 1 n 1 m reg Then a = λa a b b is a τ -factorization. If there is only one τ -factor in the factor- 1 n 1 m reg reg ization, i.e. m + n = 1, then this factor is certainly essential. If it were removed then it would ≥ imply that a were a unit, a contradiction. We now may assume that m + n 2, and therefore · · · · · · a = λa a b b is a τ -factorization implies that a is a product of regular elements and 1 n 1 m reg · · · · · · ∈ hence is regular. Moreover, we have (a) = (b b ) so ar = b b for some r R. Hence · · · · · 1 m 1 m· · · · a = λa a a r and a is regular so cancellation implies that 1 = λa a r and in particular ∈ 1 n ≤ ≤ 1 n a U(R) for all 1 i n. Hence there can be no non-unit inessential τ -divisors as desired. i ∈ · · · reg Given a τ -α-factorization of a non-unit a R, say a = λa a , we show that a is reg ≤ ≤ 1 n ≥ i essential for each(cid:98)1 i n. If n = 1, this is immediate as above. Thus n 2 and therefore ≥ ≥ a is regular for each 1 i n. Suppose for a moment that a were not essential. Then i · · · · · · · · · i ∈ (cid:98) (a) = (a a − a a a ) = (a a ). But this means there is an r R such that 1 i 1 i i+1 n 1 n · · · · · · · · · · a a − a a a = r a a . 1 i 1 i i+1 n 1 n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: τ-REGULAR

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: Let

text: R

text: be

text: a

text: r-FFR,

text: but

text: suppose

text: that

text: R

text: is

text: not

text: a

text: τ-r-FFR.

text: We

text: then

text: have

text: a

text: regular

text: element

text: a

text: Reg(R)#

text: that

text: has

text: an

text: inﬁnite

text: number

text: of

text: τ-r-factorizations

text: up

text: to

text: rearrangement

text: and

text: associate,

text: but

text: again

text: each

text: of

text: these

text: is

text: also

text: r-factorization

text: and

text: are

text: still

text: unique

text: up

text: to

text: rearrangement

text: and

text: associates

text: which

text: contradicts

text: the

text: fact

text: that

text: R

text: is

text: a

text: r-FFR.

text: Let

text: a

text: Reg(R).

text: Every

text: τ-r-divisor

text: divisor

text: is

text: a

text: regular

text: divisor

text: of

text: a,

text: so

text: there

text: can

text: be

text: only

text: ﬁnitely

text: many

text: up

text: to

text: associate.

text: Suppose

text: we

text: have

text: an

text: inﬁnite

text: sequence

text: a

text: a

text: Reg(R)#

text: for

text: all

text: k

text: with

text: a

text: a

text: but

text: i

text: k

text: τ

text: a

text: a

text: for

text: all

text: n

text: But

text: then

text: we

text: still

text: have

text: a

text: a

text: a

text: Reg(R)#

text: for

text: all

text: k

text: but

text: a

text: a

text: n

text: τ

text: n

text: k

text: so

text: we

text: contradict

text: r-ACCP.

text: Concluding

text: the

text: proof.

text: Corollary

text: The

text: r-UFRs,

text: r-FFRs,

text: r-HFRs,

text: r-BFRs

text: as

text: deﬁned

text: in

text: Section

text: satisfy

text: r-

text: ACCP,

text: and

text: therefore

text: τ-r-ACCP.

text: Hence

text: for

text: τ

text: reﬁnable,

text: each

text: is

text: τ-r-atomic

text: by

text: Theorem

text: and

text: Theorem

text: The

text: following

text: diagram

text: summarizes

text: our

text: results

text: represents

text: τ

text: being

text: reﬁnable):

text: r-BFR

text: r-ACCP

text: τ-r-UFR

text: FR

text: τ-r-BFR

text: CCP

text: τ-r-atomic

text: r-FFR

text: FFR

text: τ-ACCP

text: r-WFFR

text: τ-atomic

text: τ-r-idf

text: τ-r-idf

text: Lemma

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: let

text: τ

text: be

text: a

text: symmetric

text: relation

text: on

text: R#.

text: Let

text: α

text: atomic,

text: strongly

text: atomic,

text: m-atomic,

text: unreﬁnably

text: atomic,

text: very

text: strongly

text: atomic

text: Every

text: non-unit

text: element

text: in

text: a

text: τ

text: -U-α-factorization

text: is

text: an

text: essential

text: divisor.

text: Moreover,

text: given

text: a

text: τ

text: -α-

text: reg

text: reg

text: factorization,

text: every

text: τ-factor

text: is

text: essential.

text: When

text: α

text: we

text: mean

text: simply

text: a

text: τ

text: -U-factorization.

text: reg

text: Proof.

text: Let

text: a

text: R

text: be

text: a

text: non-unit

text: and

text: let

text: a

text: λa

text: a

text: b

text: b

text: be

text: a

text: τ

text: -U-α-factorization.

text: n

text: m

text: reg

text: Then

text: a

text: λa

text: a

text: b

text: b

text: is

text: a

text: τ

text: -factorization.

text: If

text: there

text: is

text: only

text: one

text: τ

text: -factor

text: in

text: the

text: factor-

text: n

text: m

text: reg

text: reg

text: ization,

text: i.e.

text: m

text: n

text: then

text: this

text: factor

text: is

text: certainly

text: essential.

text: If

text: it

text: were

text: removed

text: then

text: it

text: would

text: imply

text: that

text: a

text: were

text: a

text: unit,

text: a

text: contradiction.

text: We

text: now

text: may

text: assume

text: that

text: m

text: n

text: and

text: therefore

text: a

text: λa

text: a

text: b

text: b

text: is

text: a

text: τ

text: -factorization

text: implies

text: that

text: a

text: is

text: a

text: product

text: of

text: regular

text: elements

text: and

text: n

text: m

text: reg

text: hence

text: is

text: regular.

text: Moreover,

text: we

text: have

text: (a)

text: (b

text: b

text: so

text: ar

text: b

text: b

text: for

text: some

text: r

text: R.

text: Hence

text: m

text: m·

text: a

text: λa

text: a

text: a

text: r

text: and

text: a

text: is

text: regular

text: so

text: cancellation

text: implies

text: that

text: λa

text: a

text: r

text: and

text: in

text: particular

text: n

text: n

text: a

text: U(R)

text: for

text: all

text: i

text: n.

text: Hence

text: there

text: can

text: be

text: no

text: non-unit

text: inessential

text: τ

text: -divisors

text: as

text: desired.

text: i

text: reg

text: Given

text: a

text: τ

text: -α-factorization

text: of

text: a

text: non-unit

text: a

text: R,

text: say

text: a

text: λa

text: a

text: we

text: show

text: that

text: a

text: is

text: reg

text: n

text: i

text: essential

text: for

text: i

text: n.

text: If

text: n

text: this

text: is

text: immediate

text: as

text: above.

text: Thus

text: n

text: and

text: therefore

text: a

text: is

text: regular

text: for

text: each

text: i

text: n.

text: Suppose

text: for

text: a

text: moment

text: that

text: a

text: were

text: not

text: essential.

text: Then

text: i

text: i

text: (a)

text: (a

text: a

text: a

text: a

text: a

text: (a

text: a

text: But

text: this

text: means

text: there

text: is

text: an

text: r

text: R

text: such

text: that

text: i

text: i

text: n

text: n

text: a

text: a

text: a

text: a

text: a

text: r

text: a

text: a

text: i

text: i

text: n

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: τ-REGULAR FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS 3 × ⇔ ∈ (3) Let S be a non-empty subset of R# and let τ = S S, aτb a, b S. So τ is mul- tiplicative (resp. divisive) if and only if S is multiplicatively closed (resp. closed under non-unit factors). A non-trivial τ-factorization is up to unit factors a factorization into elements from S. ⇔ (4) Let (cid:63) be a star-operation on R and deﬁne aτb (a, b)(cid:63) = R, that is a and b are (cid:63)- coprime or (cid:63)-comaximal. This particular operation has been studied more in depth by Jason Juett in [15]. When (cid:63) = d, the identity star operation, we get the co-maximal factorizations of S. McAdam and R. Swan, in [16]. ⇔ ∈ (5) Let aτ b ab = 0. Then every a R# is a τ-atom. The only nontrivial τ-factorizations z · · · (cid:54) are 0 = λa . . . a where a a = 0 for all i = j. This example was studied extensively 1 n i j in [17] and has a close relationship with zero-divisor graphs. ⇔ ∈ (6) Let aτb a, b Reg(R). Then this gives us the regular factorization studied in [6]. This is the inspiration for Section . ⊆ × ∩ × (7) Let τ R# R#, then we deﬁne τ := τ (Reg(R) Reg(R)). Because the collection reg of regular elements is a saturated, multiplicatively closed set, this has the eﬀect of only allowing trivial factorizations of the zero-divisors. This is the type of τ-factorization we would like to use to compare with the notion of τ-factorizations by way of the regular factorizations studied in [6]. This will be studied more in depth in Section . ∈ We now summarize several of the deﬁnitions given in [17] and [19]. Let a R be a non-unit. · · · Then a is said to be τ-irreducible or τ-atomic if for any τ-factorization a = λa a , we ∼ 1 n have a a for some i. We will say a is τ-strongly irreducible or τ-strongly atomic if for any i · · · ≈ τ-factorization a = λa a , we have a a for some a . We will say that a is τ-m-irreducible 1 n i · · · i ∼ or τ-m-atomic if for any τ-factorization a = λa a , we have a a for all i. Note: the 1 n i m is for “maximal” since such an a is maximal among principal ideals generated by elements ∈ which occur as τ-factors of a. As in [19], a R is said to be a τ-unreﬁnable atom if a admits only trivial ∼τ-factorizations. 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 [17] and [19] for more equivalent deﬁnitions of these various forms of τ-irreducibility. We have the following relationship between the various types of τ-irreducibles which is proved in [17, Theorem 3.9] as well as [19]. Theorem 2.2. Let R be a commutative ring with 1 and τ be a symmetric relation on R#. Let ∈ a R be a non-unit. The following diagram illustrates the relationship between the various ≈ types of τ-irreducibles a might satisfy where represents R being a strongly associate ring. (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) τ-very strongly irred. τ-unreﬁnably irred. τ-strong(cid:75)(cid:83)ly irred. τ(cid:50)(cid:58) -irred. ≈ (cid:38)(cid:46) τ-m-irred. Following A. Bouvier, a ring R is said to be pr´esimpliﬁable if x = xy implies x = 0 or ∈ y U(R) as in [9, 10, 11, 12]. When R is pr´esimpliﬁable, the various associate relations coin- cide. As seen in [17], for non-zero elements, if R is pr´esimpliﬁable, then τ-irreducible will imply τ-very strongly irreducible and the various types of irreducible elements will also coincide. Any integral domain or quasi-local ring is pr´esimpliﬁable. Examples are given in [4] and abound in the literature which show that in a general commutative ring setting, each of these types of irreducible elements are distinct. For further discussion of the diﬀerent τ-irreducible elements, the reader is directed to [17]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: τ-REGULAR

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: Let

text: S

text: be

text: a

text: non-empty

text: subset

text: of

text: R#

text: and

text: let

text: τ

text: S

text: S,

text: aτb

text: a,

text: b

text: S.

text: So

text: τ

text: is

text: mul-

text: tiplicative

text: (resp.

text: divisive)

text: if

text: and

text: only

text: if

text: S

text: is

text: multiplicatively

text: closed

text: (resp.

text: closed

text: under

text: non-unit

text: factors).

text: A

text: non-trivial

text: τ-factorization

text: is

text: up

text: to

text: unit

text: factors

text: a

text: factorization

text: into

text: elements

text: from

text: S.

text: Let

text: be

text: a

text: star-operation

text: on

text: R

text: and

text: deﬁne

text: aτb

text: (a,

text: R,

text: that

text: is

text: a

text: and

text: b

text: are

text: coprime

text: or

text: This

text: particular

text: operation

text: has

text: been

text: studied

text: more

text: in

text: depth

text: by

text: Jason

text: Juett

text: in

text: When

text: d,

text: the

text: identity

text: star

text: operation,

text: we

text: get

text: the

text: co-maximal

text: factorizations

text: of

text: S.

text: McAdam

text: and

text: R.

text: Swan,

text: in

text: Let

text: aτ

text: b

text: ab

text: Then

text: every

text: a

text: R#

text: is

text: a

text: τ-atom.

text: The

text: only

text: nontrivial

text: τ-factorizations

text: z

text: are

text: λa

text: a

text: where

text: a

text: a

text: for

text: all

text: i

text: j.

text: This

text: example

text: was

text: studied

text: extensively

text: n

text: i

text: j

text: in

text: and

text: has

text: a

text: close

text: relationship

text: with

text: zero-divisor

text: graphs.

text: Let

text: aτb

text: a,

text: b

text: Reg(R).

text: Then

text: this

text: gives

text: us

text: the

text: regular

text: factorization

text: studied

text: in

text: This

text: is

text: the

text: inspiration

text: for

text: Section

text: Let

text: τ

text: R#

text: R#,

text: then

text: we

text: deﬁne

text: τ

text: τ

text: (Reg(R)

text: Reg(R)).

text: Because

text: the

text: collection

text: reg

text: of

text: regular

text: elements

text: is

text: a

text: saturated,

text: multiplicatively

text: closed

text: set,

text: this

text: has

text: the

text: eﬀect

text: of

text: only

text: allowing

text: trivial

text: factorizations

text: of

text: the

text: zero-divisors.

text: This

text: is

text: the

text: type

text: of

text: τ-factorization

text: we

text: would

text: like

text: to

text: use

text: to

text: compare

text: with

text: the

text: notion

text: of

text: τ-factorizations

text: by

text: way

text: of

text: the

text: regular

text: factorizations

text: studied

text: in

text: This

text: will

text: be

text: studied

text: more

text: in

text: depth

text: in

text: Section

text: We

text: now

text: summarize

text: several

text: of

text: the

text: deﬁnitions

text: given

text: in

text: and

text: Let

text: a

text: R

text: be

text: a

text: non-unit.

text: Then

text: a

text: is

text: said

text: to

text: be

text: τ-irreducible

text: or

text: τ-atomic

text: if

text: for

text: any

text: τ-factorization

text: a

text: λa

text: a

text: we

text: n

text: have

text: a

text: a

text: for

text: some

text: i.

text: We

text: will

text: say

text: a

text: is

text: τ-strongly

text: irreducible

text: or

text: τ-strongly

text: atomic

text: if

text: for

text: any

text: i

text: τ-factorization

text: a

text: λa

text: a

text: we

text: have

text: a

text: a

text: for

text: some

text: a

text: We

text: will

text: say

text: that

text: a

text: is

text: τ-m-irreducible

text: n

text: i

text: i

text: or

text: τ-m-atomic

text: if

text: for

text: any

text: τ-factorization

text: a

text: λa

text: a

text: we

text: have

text: a

text: a

text: for

text: all

text: i.

text: Note:

text: the

text: n

text: i

text: m

text: is

text: for

text: “maximal”

text: since

text: such

text: an

text: a

text: is

text: maximal

text: among

text: principal

text: ideals

text: generated

text: by

text: elements

text: which

text: occur

text: as

text: τ-factors

text: of

text: a.

text: As

text: in

text: a

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-unreﬁnable

text: atom

text: if

text: a

text: admits

text: only

text: trivial

text: ∼τ-factorizations.

text: We

text: will

text: say

text: that

text: a

text: is

text: τ-very

text: strongly

text: irreducible

text: or

text: τ-very

text: strongly

text: atomic

text: if

text: a

text: a

text: and

text: a

text: has

text: no

text: non-trivial

text: τ-factorizations.

text: See

text: and

text: for

text: more

text: equivalent

text: deﬁnitions

text: of

text: these

text: various

text: forms

text: of

text: τ-irreducibility.

text: We

text: have

text: the

text: following

text: relationship

text: between

text: the

text: various

text: types

text: of

text: τ-irreducibles

text: which

text: is

text: proved

text: in

text: Theorem

text: as

text: well

text: as

text: Theorem

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: τ

text: be

text: a

text: symmetric

text: relation

text: on

text: R#.

text: Let

text: a

text: R

text: be

text: a

text: non-unit.

text: The

text: following

text: diagram

text: illustrates

text: the

text: relationship

text: between

text: the

text: various

text: types

text: of

text: τ-irreducibles

text: a

text: might

text: satisfy

text: where

text: represents

text: R

text: being

text: a

text: strongly

text: associate

text: ring.

text: τ-very

text: strongly

text: irred.

text: τ-unreﬁnably

text: irred.

text: irred.

text: -irred.

text: τ-m-irred.

text: Following

text: A.

text: Bouvier,

text: a

text: ring

text: R

text: is

text: said

text: to

text: be

text: pr´esimpliﬁable

text: if

text: x

text: xy

text: implies

text: x

text: or

text: y

text: U(R)

text: as

text: in

text: When

text: R

text: is

text: pr´esimpliﬁable,

text: the

text: various

text: associate

text: relations

text: coin-

text: cide.

text: As

text: seen

text: in

text: for

text: non-zero

text: elements,

text: if

text: R

text: is

text: pr´esimpliﬁable,

text: then

text: τ-irreducible

text: will

text: imply

text: τ-very

text: strongly

text: irreducible

text: and

text: the

text: various

text: types

text: of

text: irreducible

text: elements

text: will

text: also

text: coincide.

text: Any

text: integral

text: domain

text: or

text: quasi-local

text: ring

text: is

text: pr´esimpliﬁable.

text: Examples

text: are

text: given

text: in

text: and

text: abound

text: in

text: the

text: literature

text: which

text: show

text: that

text: in

text: a

text: general

text: commutative

text: ring

text: setting,

text: each

text: of

text: these

text: types

text: of

text: irreducible

text: elements

text: are

text: distinct.

text: For

text: further

text: discussion

text: of

text: the

text: diﬀerent

text: τ-irreducible

text: elements,

text: the

text: reader

text: is

text: directed

text: to

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text: τ-REGULAR FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS 5 · · · · · · which λa a b b is also a τ-factorization. 1 n 1 m Given a symmetric relation τ on R#, we say R is τ-U-reﬁnable if for every τ-U-factorization of ∈ · · · (cid:100) · · · (cid:101) any non-unit a U(R), a = λa a b b , any τ-U-factorization of an essential divisors, (cid:48) · · · (cid:100) · · · (cid:101) 1 n 1 m b = λ c c (cid:48) d d (cid:48) satisﬁes i 1 n 1 m (cid:48) · · · · · · (cid:100) · · · · · · · · · (cid:101) a = λλ a a c c (cid:48) b b − d d (cid:48)b b 1 n 1 n 1 i 1 1 m i+1 m is a τ-U-factorization. ∈ { } Let α irreducible, strongly irreducible, m-irreducible, very strongly irreducible . Let a be · · · (cid:100) · · · (cid:101) a non-unit. If a = λa a a b b b is a τ-U-factorization, then this factorization is said 1 2 n 1 2 m to be a τ-U-α-factorization if it is a τ-U-factorization and the essential divisors b are τ-α for ≤ ≤ i 1 i m. We now deﬁne the ﬁnite factorization properties using the τ-U-factorization approach. Let ∈ { α irreducible, strongly irreducible, m-irreducible, unreﬁnably 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 (cid:40) (cid:40) · · · principal ideals (a ) (a ) such that a is an essential divisor in some τ-U-factorization 1 2 i+1 of a , for each i terminates after ﬁnitely many principal ideals. R is said to be a τ-U-BFR if for i ∈ 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-β-UFR if R is a τ-U-α-HFR and the essential divisors of any two τ-U-α-factorizations can be rearranged to match up to β. The following diagram summarizes the main results from from [18, Theorem 4.3 and Theorem ≈ † 4.4] where represents R being strongly associate, and represents R is τ-U-reﬁnable: ≈ (cid:107)(cid:115) τ-α-β-UFR τ-U(cid:49)(cid:57) -α-HFR τ-α-HFR † ≈ (cid:11)(cid:19) † (cid:36)(cid:44) † † (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) τ-U-α-β-UFR τ-U-β-F(cid:93)(cid:101) FR τ-U-B(cid:75)(cid:83) FR τ-U-A(cid:75)(cid:83) CCP τ-U(cid:75)(cid:83) -α (cid:11)(cid:19) − (cid:43)(cid:51) τ-β-WFFR τ-U-β-WFFR τ-BFR τ ACCP τ-α (cid:11)(cid:19) (cid:43)(cid:51) τ-α-β df ring τ-U-α-β df ring τ-β-FFR ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: τ-REGULAR

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: which

text: λa

text: a

text: b

text: b

text: is

text: also

text: a

text: τ-factorization.

text: n

text: m

text: Given

text: a

text: symmetric

text: relation

text: τ

text: on

text: R#,

text: we

text: say

text: R

text: is

text: τ-U-reﬁnable

text: if

text: for

text: every

text: τ-U-factorization

text: of

text: any

text: non-unit

text: a

text: U(R),

text: a

text: λa

text: a

text: b

text: b

text: any

text: τ-U-factorization

text: of

text: an

text: essential

text: divisors,

text: n

text: m

text: b

text: λ

text: c

text: c

text: d

text: d

text: satisﬁes

text: i

text: n

text: m

text: a

text: λλ

text: a

text: a

text: c

text: c

text: b

text: b

text: d

text: d

text: b

text: n

text: n

text: i

text: m

text: m

text: is

text: a

text: τ-U-factorization.

text: Let

text: α

text: irreducible,

text: strongly

text: irreducible,

text: m-irreducible,

text: very

text: strongly

text: irreducible

text: Let

text: a

text: be

text: a

text: non-unit.

text: If

text: a

text: λa

text: a

text: a

text: b

text: b

text: b

text: is

text: a

text: τ-U-factorization,

text: then

text: this

text: factorization

text: is

text: said

text: n

text: m

text: to

text: be

text: a

text: τ-U-α-factorization

text: if

text: it

text: is

text: a

text: τ-U-factorization

text: and

text: the

text: essential

text: divisors

text: b

text: are

text: τ-α

text: for

text: i

text: i

text: m.

text: We

text: now

text: deﬁne

text: the

text: ﬁnite

text: factorization

text: properties

text: using

text: the

text: τ-U-factorization

text: approach.

text: Let

text: α

text: irreducible,

text: strongly

text: irreducible,

text: m-irreducible,

text: unreﬁnably

text: irreducible,

text: very

text: strongly

text: irreducible

text: and

text: let

text: β

text: associate,

text: strongly

text: associate,

text: very

text: strongly

text: associate

text: R

text: is

text: said

text: to

text: be

text: τ-U-α

text: if

text: for

text: all

text: non-units

text: a

text: R,

text: there

text: is

text: a

text: τ-U-α-factorization

text: of

text: a.

text: R

text: is

text: said

text: to

text: satisfy

text: τ-U-ACCP

text: (ascending

text: chain

text: condition

text: on

text: principal

text: ideals)

text: if

text: every

text: properly

text: ascending

text: chain

text: of

text: principal

text: ideals

text: (a

text: (a

text: such

text: that

text: a

text: is

text: an

text: essential

text: divisor

text: in

text: some

text: τ-U-factorization

text: of

text: a

text: for

text: each

text: i

text: terminates

text: after

text: ﬁnitely

text: many

text: principal

text: ideals.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-U-BFR

text: if

text: for

text: i

text: all

text: non-units

text: a

text: R,

text: there

text: is

text: a

text: bound

text: on

text: the

text: number

text: of

text: essential

text: divisors

text: in

text: any

text: τ-U-factorization

text: of

text: a.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-U-β-FFR

text: if

text: for

text: all

text: non-units

text: a

text: R,

text: there

text: are

text: only

text: ﬁnitely

text: many

text: τ-U-

text: factorizations

text: up

text: to

text: rearrangement

text: of

text: the

text: essential

text: divisors

text: and

text: β.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-U-β-

text: WFFR

text: if

text: for

text: all

text: non-units

text: a

text: R,

text: there

text: are

text: only

text: ﬁnitely

text: many

text: essential

text: divisors

text: among

text: all

text: τ-U-factorizations

text: of

text: a

text: up

text: to

text: β.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-U-α-β-divisor

text: ﬁnite

text: (df)

text: ring

text: if

text: for

text: all

text: non-

text: units

text: a

text: R,

text: there

text: are

text: only

text: ﬁnitely

text: many

text: essential

text: τ-α

text: divisors

text: up

text: to

text: β

text: in

text: the

text: τ-U-factorizations

text: of

text: a.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-U-α-HFR

text: if

text: R

text: is

text: τ-U-α

text: and

text: for

text: all

text: non-units

text: a

text: R,

text: the

text: number

text: of

text: essential

text: divisors

text: in

text: any

text: τ-U-α-factorization

text: of

text: a

text: is

text: the

text: same.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-U-α-β-UFR

text: if

text: R

text: is

text: a

text: τ-U-α-HFR

text: and

text: the

text: essential

text: divisors

text: of

text: any

text: two

text: τ-U-α-factorizations

text: can

text: be

text: rearranged

text: to

text: match

text: up

text: to

text: β.

text: The

text: following

text: diagram

text: summarizes

text: the

text: main

text: results

text: from

text: from

text: Theorem

text: and

text: Theorem

text: where

text: represents

text: R

text: being

text: strongly

text: associate,

text: and

text: represents

text: R

text: is

text: τ-U-reﬁnable:

text: τ-α-β-UFR

text: -α-HFR

text: τ-α-HFR

text: τ-U-α-β-UFR

text: FR

text: FR

text: CCP

text: -α

text: τ-β-WFFR

text: τ-U-β-WFFR

text: τ-BFR

text: τ

text: ACCP

text: τ-α

text: τ-α-β

text: df

text: ring

text: τ-U-α-β

text: df

text: ring

text: τ-β-FFR

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: τ-REGULAR FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS 7 } τ-atomic-factorizations . Then it is clear that R is a τ-r-HFR if and only if R is τ-atomic and τ-r-ρ(R)=1. ∈ R is said to be a τ-r-bounded factorization domain (BFR) if for every a Reg(R) there exists · · · ≤ a natural number N (a) such that for all τ-r-factorizations a = λa a , we have n N (a). r 1 ∈ n r R is said to be a τ-r-irreducible-divisor-ﬁnite ring (idf ring) if each a Reg(R)# has at most a ﬁnite number of non-associate τ-irreducible τ-divisors. R is said to be a τ-r-ﬁnite factorization ∈ ring (FFR) if for every a Reg(R)#, a has only a ﬁnite number (up to order and associates) of τ-factorizations. R is said to be a τ-r-weak ﬁnite factorization ring (WFFR) if for every ∈ a Reg(R)# there are only a ﬁnite number of non-associate τ-divisors. Proposition 3.2. Let R be a commutative ring with 1. Let τ be a symmetric relation on R# with τ reﬁnable, then the following are equivalent. (1) R is a τ-r-FFR. (2) R is a τ-r-WFFR. (3) R is a τ-r-atomic τ-r-idf ring. ∈ (4) R is τ-r-atomic and each a Reg(R)#, a has only ﬁnitely many τ-r-atomic τ-factorizations up to order and associates. ∈ ∈ (5) For all a Reg(R)#, there are only ﬁnitely many b Reg(R)# up to associate such that b occurs as a τ-factor in a τ-r-factorization of a. ∈ ∈ (6) For all a Reg(R)#, (a) is contained in only ﬁnitely many principal ideals (b) where b Reg(R)# such that b occurs as a τ-factor in a τ-r-factorization of a. ∈ ∈ (7) For all a Reg(R)#, there are only ﬁnitely many b Reg(R)# up to associate such that | b a. τ ∈ ∈ (8) For all a Reg(R)#, (a) is contained in only ﬁnitely many principal ideals (b) where b | Reg(R)# such that b a. τ ⇒ ∈ Proof. (1) (2) Let R be a τ-r-FFR and a Reg(R)#, then there are only a ﬁnite number of τ-factorizations (up to order and associate), each of these is of ﬁnite length. Hence, since every τ-divisor of a must be among these up to associate, R is a τ-r-WFFR. ⇒ ∈ (2) (3) Let R be a τ-r-WFFR and a Reg(R)#. If a has a ﬁnite number of τ-divisors, then certainly it has a ﬁnite number of irreducible τ-divisors, so it suﬃces to show a has a τ-r- atomic factorization. We instead show the stronger condition, that R satisﬁes τ-r-ACCP, that (cid:40) (cid:40) · · · ⊂ (cid:40) · · · is any chain of principal ideals generated by regular elements (a ) (a ) (a ) 0 ∈ 1 i with a occurring as a τ-factor in a τ-r-factorization of a and a Reg(R)# for all i comes i+1 i i to a halt. Suppose there is an inﬁnite chain, but then each a is a τ-divisor of a and none of i 0 them are associate since each containment is proper, so we would have an inﬁnite number of non-associate τ-r-divisors contradicting the fact that R is a τ-r-WFFR (note: we use strongly here that τ is reﬁnable to ensure that at each step we retain a τ-factorization). ⇒ (3) (1) This proof is similar to [1, Thm 5.1]. Let R be a τ-r-atomic τ-r-idf ring and ∈ · · · x Reg(R)#. Let x , , x be the τ-r-irreducible τ-factors of x, in particular they are all 1 n · · · regular elements of R. Suppose that in a τ-factorization of x, x = λxs1 xsn, we always have ≤ ≤ ≤ ≤ 1 n 0 s N for each 1 i n. Then there is a bound on the number of non-associate factors i i of x. So we suppose that this is not the case. There must then be some s which is not bounded, ≥ i · · · we assume it is the ﬁrst one s . Hence for each k 1, we can write x = λ xsk1 xskn, where ∈ 1 · · · k 1{ } n λ U(R) and s < s < s < . Suppose that in this set of factorizations s is bounded k 11 21 31 ki

text: τ-REGULAR

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: τ-atomic-factorizations

text: Then

text: it

text: is

text: clear

text: that

text: R

text: is

text: a

text: τ-r-HFR

text: if

text: and

text: only

text: if

text: R

text: is

text: τ-atomic

text: and

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-r-bounded

text: factorization

text: domain

text: (BFR)

text: if

text: for

text: every

text: a

text: Reg(R)

text: there

text: exists

text: a

text: natural

text: number

text: N

text: (a)

text: such

text: that

text: for

text: all

text: τ-r-factorizations

text: a

text: λa

text: a

text: we

text: have

text: n

text: N

text: (a).

text: r

text: n

text: r

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-r-irreducible-divisor-ﬁnite

text: ring

text: (idf

text: ring)

text: if

text: each

text: a

text: Reg(R)#

text: has

text: at

text: most

text: a

text: ﬁnite

text: number

text: of

text: non-associate

text: τ-irreducible

text: τ-divisors.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-r-ﬁnite

text: factorization

text: ring

text: (FFR)

text: if

text: for

text: every

text: a

text: Reg(R)#,

text: a

text: has

text: only

text: a

text: ﬁnite

text: number

text: (up

text: to

text: order

text: and

text: associates)

text: of

text: τ-factorizations.

text: R

text: is

text: said

text: to

text: be

text: a

text: τ-r-weak

text: ﬁnite

text: factorization

text: ring

text: (WFFR)

text: if

text: for

text: every

text: a

text: Reg(R)#

text: there

text: are

text: only

text: a

text: ﬁnite

text: number

text: of

text: non-associate

text: τ-divisors.

text: Proposition

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: Let

text: τ

text: be

text: a

text: symmetric

text: relation

text: on

text: R#

text: with

text: τ

text: reﬁnable,

text: then

text: the

text: following

text: are

text: equivalent.

text: R

text: is

text: a

text: τ-r-FFR.

text: R

text: is

text: a

text: τ-r-WFFR.

text: R

text: is

text: a

text: τ-r-atomic

text: τ-r-idf

text: ring.

text: R

text: is

text: τ-r-atomic

text: and

text: each

text: a

text: Reg(R)#,

text: a

text: has

text: only

text: ﬁnitely

text: many

text: τ-r-atomic

text: τ-factorizations

text: up

text: to

text: order

text: and

text: associates.

text: For

text: all

text: a

text: Reg(R)#,

text: there

text: are

text: only

text: ﬁnitely

text: many

text: b

text: Reg(R)#

text: up

text: to

text: associate

text: such

text: that

text: b

text: occurs

text: as

text: a

text: τ-factor

text: in

text: a

text: τ-r-factorization

text: of

text: a.

text: For

text: all

text: a

text: Reg(R)#,

text: (a)

text: is

text: contained

text: in

text: only

text: ﬁnitely

text: many

text: principal

text: ideals

text: (b)

text: where

text: b

text: Reg(R)#

text: such

text: that

text: b

text: occurs

text: as

text: a

text: τ-factor

text: in

text: a

text: τ-r-factorization

text: of

text: a.

text: For

text: all

text: a

text: Reg(R)#,

text: there

text: are

text: only

text: ﬁnitely

text: many

text: b

text: Reg(R)#

text: up

text: to

text: associate

text: such

text: that

text: b

text: a.

text: τ

text: For

text: all

text: a

text: Reg(R)#,

text: (a)

text: is

text: contained

text: in

text: only

text: ﬁnitely

text: many

text: principal

text: ideals

text: (b)

text: where

text: b

text: Reg(R)#

text: such

text: that

text: b

text: a.

text: τ

text: Proof.

text: Let

text: R

text: be

text: a

text: τ-r-FFR

text: and

text: a

text: Reg(R)#,

text: then

text: there

text: are

text: only

text: a

text: ﬁnite

text: number

text: of

text: τ-factorizations

text: (up

text: to

text: order

text: and

text: associate),

text: each

text: of

text: these

text: is

text: of

text: ﬁnite

text: length.

text: Hence,

text: since

text: every

text: τ-divisor

text: of

text: a

text: must

text: be

text: among

text: these

text: up

text: to

text: associate,

text: R

text: is

text: a

text: τ-r-WFFR.

text: Let

text: R

text: be

text: a

text: τ-r-WFFR

text: and

text: a

text: Reg(R)#.

text: If

text: a

text: has

text: a

text: ﬁnite

text: number

text: of

text: τ-divisors,

text: then

text: certainly

text: it

text: has

text: a

text: ﬁnite

text: number

text: of

text: irreducible

text: τ-divisors,

text: so

text: it

text: suﬃces

text: to

text: show

text: a

text: has

text: a

text: τ-r-

text: atomic

text: factorization.

text: We

text: instead

text: show

text: the

text: stronger

text: condition,

text: that

text: R

text: satisﬁes

text: τ-r-ACCP,

text: that

text: is

text: any

text: chain

text: of

text: principal

text: ideals

text: generated

text: by

text: regular

text: elements

text: (a

text: (a

text: (a

text: i

text: with

text: a

text: occurring

text: as

text: a

text: τ-factor

text: in

text: a

text: τ-r-factorization

text: of

text: a

text: and

text: a

text: Reg(R)#

text: for

text: all

text: i

text: comes

text: i

text: i

text: to

text: a

text: halt.

text: Suppose

text: there

text: is

text: an

text: inﬁnite

text: chain,

text: but

text: then

text: each

text: a

text: is

text: a

text: τ-divisor

text: of

text: a

text: and

text: of

text: i

text: them

text: are

text: associate

text: since

text: each

text: containment

text: is

text: proper,

text: so

text: we

text: would

text: have

text: an

text: inﬁnite

text: number

text: of

text: non-associate

text: τ-r-divisors

text: contradicting

text: the

text: fact

text: that

text: R

text: is

text: a

text: τ-r-WFFR

text: (note:

text: we

text: use

text: strongly

text: here

text: that

text: τ

text: is

text: reﬁnable

text: to

text: ensure

text: that

text: at

text: each

text: step

text: we

text: retain

text: a

text: τ-factorization).

text: This

text: proof

text: is

text: similar

text: to

text: Thm

text: Let

text: R

text: be

text: a

text: τ-r-atomic

text: τ-r-idf

text: ring

text: and

text: x

text: Reg(R)#.

text: Let

text: x

text: x

text: be

text: the

text: τ-r-irreducible

text: τ-factors

text: of

text: x,

text: in

text: particular

text: they

text: are

text: all

text: n

text: regular

text: elements

text: of

text: R.

text: Suppose

text: that

text: in

text: a

text: τ-factorization

text: of

text: x,

text: x

text: xsn,

text: we

text: always

text: have

text: n

text: s

text: N

text: for

text: each

text: i

text: n.

text: Then

text: there

text: is

text: a

text: bound

text: on

text: the

text: number

text: of

text: non-associate

text: factors

text: i

text: i

text: of

text: x.

text: So

text: we

text: suppose

text: that

text: this

text: is

text: not

text: the

text: case.

text: There

text: must

text: then

text: be

text: some

text: s

text: which

text: is

text: not

text: bounded,

text: i

text: we

text: assume

text: it

text: is

text: the

text: ﬁrst

text: one

text: s

text: Hence

text: for

text: each

text: k

text: we

text: can

text: write

text: x

text: λ

text: xskn,

text: where

text: k

text: n

text: λ

text: U(R)

text: and

text: s

text: s

text: s

text: Suppose

text: that

text: in

text: this

text: set

text: of

text: factorizations

text: s

text: is

text: bounded

text: k

text: ki


text: 8 CHRISTOPHER PARK MOONEY ≤ · · · for each i with 1 < i n. Then since there are only ﬁnitely many choices for s , , s we must have s = s , · · · , s = s for some j > k. But then λ xsj1 · · · xsjn = x = λk2xsk1 · · ·kxnskn, k2 j2 kn jn j 1 n k 1 n but since each x is regular, we can cancel to get λ xsj1 = λ xsk1, where s > s , but then x i j 1 k 1 j1 k1 1 would be a unit, a contradiction. { } ≤ Thus we must have some set s for a ﬁxed i with 1 < i n is unbounded, say for ki · · · i = 2. By taking subsequences at each stage, we may assume that s < s < s < and · · · 11 21 ≤31 ≤ s < s < s < . Continuing in this manner, we may assume for each 1 i n that s12 < s22 < s32 < · · · . But then we would have λ xs11 · · · xs1n = x = λ xs21 · · · xs2n where 1i 2i 3i 1 1 n 2 1 n s < s , a contradiction as again, we would have x must be units after cancellation, which is 1i 2i i impossible. ⇒ (1) (4) This is clear as we have already seen that a τ-r-FFR is τ-r-atomic and a τ-r-atomic factorization is certainly a τ-r-factorization, so there must be a ﬁnite number of τ-r-atomic ∈ factorizations up to order and associate for every a Reg(R)#. ⇒ ∈ (4) (3) Let a Reg(R)#, then there are a ﬁnite number of τ-r-atomic factorizations, each has a ﬁnite number of τ-r-atomic factors, so the collection of τ-r-atomic divisors is ﬁnite, so R is a τ-r-atomic τ-r-idf ring. (5), (6) are restatements of (2) and their equivalence is immediate. Furthermore, (5) and (7) ∈ | (resp. (6) and (8)) are seen to be equivalent after noting that for b Reg(R), a b implies there · · · τ is some τ-factorization b = λaa a , but since b is regular and the set of regular elements is 1 n (cid:3) saturated, every τ-factor must be regular so this is really a τ-factorization. Theorem 3.3. Let R be a commutative ring with 1, with τ a symmetric relation on R#. We have the following. (1) R is a τ-r-UFR implies R is a τ-r-HFR. (2) For τ reﬁnable, R is a τ-r-HFR implies R is a τ-r-BFR. (3) For τ reﬁnable, R is a τ-r-UFR implies R is a τ-r-FFR. (4) R is a τ-r-FFR implies R is a τ-r-BFR. (5) For τ reﬁnable, R is a τ-r-BFR implies R satisﬁes τ-r-ACCP. (6) For τ reﬁnable, R satisﬁes τ-r-ACCP implies R is τ-r-atomic. Proof. (1) This is immediate from the deﬁnition. · · · (2) Let R be a τ-r-HFR. Suppose a = λa a is a τ-r-atomic factorization. We claim · · · 1 n N (a) = n. Let a = µb b be a τ-r-factorization of a. Since R is τ-r-atomic, we can ﬁnd r 1 m ≤ ≤ τ-r-atomic factorizations for b for each 1 i m. We have assumed τ to be reﬁnable, so we i can replace each b with the corresponding τ-r-atomic factorization and collect the units in the i front of the factorization and retain a τ-r-factorization which is τ-atomic and thus must have length n. The reﬁnement process can only increase the length of the factorization, so the length of the original factorization is no longer than n, proving the claim. (3) We show for τ-reﬁnable, R a τ-r-UFR, R is a τ-r-atomic τ-r-idf-ring which has been shown in Theorem 3.2 to be equivalent to being a τ-r-FFR. R being τ-r-factorial gives us τ-r- ∈ atomic for free. Furthermore, any τ-atomic factorization of a Reg(R)# has the same length, say n and can be reordered so that the associates match up. This tells us there are precisely n τ-irreducible divisors of a up to associate, hence R is a τ-r-idf-ring. ∈ (4) Suppose R is a τ-r-FFR, by deﬁnition, we know R is τ-r-atomic. Now, let a Reg(R)#, | | · · · let S be the ﬁnite set of all τ-atomic factors of a. Set N(a) = S . Let a = λa a be a ∈ { } ⊆ 1 n τ-atomic factorization of a, then a S for all i, but then a n S and hence is ﬁnite and i i i=1

text: CHRISTOPHER

text: PARK

text: MOONEY

text: for

text: each

text: i

text: with

text: i

text: n.

text: Then

text: since

text: there

text: are

text: only

text: ﬁnitely

text: many

text: choices

text: for

text: s

text: s

text: we

text: must

text: have

text: s

text: s

text: s

text: s

text: for

text: some

text: j

text: k.

text: But

text: then

text: λ

text: xsjn

text: x

text: ·kxnskn,

text: kn

text: jn

text: j

text: n

text: k

text: n

text: but

text: since

text: each

text: x

text: is

text: regular,

text: we

text: can

text: cancel

text: to

text: get

text: λ

text: λ

text: where

text: s

text: s

text: but

text: then

text: x

text: i

text: j

text: k

text: would

text: be

text: a

text: unit,

text: a

text: contradiction.

text: Thus

text: we

text: must

text: have

text: some

text: set

text: s

text: for

text: a

text: ﬁxed

text: i

text: with

text: i

text: n

text: is

text: unbounded,

text: say

text: for

text: ki

text: i

text: By

text: taking

text: subsequences

text: at

text: each

text: stage,

text: we

text: may

text: assume

text: that

text: s

text: s

text: s

text: and

text: s

text: s

text: s

text: Continuing

text: in

text: this

text: manner,

text: we

text: may

text: assume

text: for

text: each

text: i

text: n

text: that

text: But

text: then

text: we

text: would

text: have

text: λ

text: x

text: λ

text: where

text: n

text: n

text: s

text: s

text: a

text: contradiction

text: as

text: again,

text: we

text: would

text: have

text: x

text: must

text: be

text: units

text: after

text: cancellation,

text: which

text: is

text: i

text: impossible.

text: This

text: is

text: clear

text: as

text: we

text: have

text: already

text: seen

text: that

text: a

text: τ-r-FFR

text: is

text: τ-r-atomic

text: and

text: a

text: τ-r-atomic

text: factorization

text: is

text: certainly

text: a

text: τ-r-factorization,

text: so

text: there

text: must

text: be

text: a

text: ﬁnite

text: number

text: of

text: τ-r-atomic

text: factorizations

text: up

text: to

text: order

text: and

text: associate

text: for

text: every

text: a

text: Reg(R)#.

text: Let

text: a

text: Reg(R)#,

text: then

text: there

text: are

text: a

text: ﬁnite

text: number

text: of

text: τ-r-atomic

text: factorizations,

text: each

text: has

text: a

text: ﬁnite

text: number

text: of

text: τ-r-atomic

text: factors,

text: so

text: the

text: collection

text: of

text: τ-r-atomic

text: divisors

text: is

text: ﬁnite,

text: so

text: R

text: is

text: a

text: τ-r-atomic

text: τ-r-idf

text: ring.

text: are

text: restatements

text: of

text: and

text: their

text: equivalence

text: is

text: immediate.

text: Furthermore,

text: and

text: (resp.

text: and

text: are

text: seen

text: to

text: be

text: equivalent

text: after

text: noting

text: that

text: for

text: b

text: Reg(R),

text: a

text: b

text: implies

text: there

text: τ

text: is

text: some

text: τ-factorization

text: b

text: λaa

text: a

text: but

text: since

text: b

text: is

text: regular

text: and

text: the

text: set

text: of

text: regular

text: elements

text: is

text: n

text: saturated,

text: every

text: τ-factor

text: must

text: be

text: regular

text: so

text: this

text: is

text: really

text: a

text: τ-factorization.

text: Theorem

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: with

text: τ

text: a

text: symmetric

text: relation

text: on

text: R#.

text: We

text: have

text: the

text: following.

text: R

text: is

text: a

text: τ-r-UFR

text: implies

text: R

text: is

text: a

text: τ-r-HFR.

text: For

text: τ

text: reﬁnable,

text: R

text: is

text: a

text: τ-r-HFR

text: implies

text: R

text: is

text: a

text: τ-r-BFR.

text: For

text: τ

text: reﬁnable,

text: R

text: is

text: a

text: τ-r-UFR

text: implies

text: R

text: is

text: a

text: τ-r-FFR.

text: R

text: is

text: a

text: τ-r-FFR

text: implies

text: R

text: is

text: a

text: τ-r-BFR.

text: For

text: τ

text: reﬁnable,

text: R

text: is

text: a

text: τ-r-BFR

text: implies

text: R

text: satisﬁes

text: τ-r-ACCP.

text: For

text: τ

text: reﬁnable,

text: R

text: satisﬁes

text: τ-r-ACCP

text: implies

text: R

text: is

text: τ-r-atomic.

text: Proof.

text: This

text: is

text: immediate

text: from

text: the

text: deﬁnition.

text: Let

text: R

text: be

text: a

text: τ-r-HFR.

text: Suppose

text: a

text: λa

text: a

text: is

text: a

text: τ-r-atomic

text: factorization.

text: We

text: claim

text: n

text: N

text: (a)

text: n.

text: Let

text: a

text: µb

text: b

text: be

text: a

text: τ-r-factorization

text: of

text: a.

text: Since

text: R

text: is

text: τ-r-atomic,

text: we

text: can

text: ﬁnd

text: r

text: m

text: τ-r-atomic

text: factorizations

text: for

text: b

text: for

text: each

text: i

text: m.

text: We

text: have

text: assumed

text: τ

text: to

text: be

text: reﬁnable,

text: so

text: we

text: i

text: can

text: replace

text: each

text: b

text: with

text: the

text: corresponding

text: τ-r-atomic

text: factorization

text: and

text: collect

text: the

text: units

text: in

text: the

text: i

text: front

text: of

text: the

text: factorization

text: and

text: retain

text: a

text: τ-r-factorization

text: which

text: is

text: τ-atomic

text: and

text: thus

text: must

text: have

text: length

text: n.

text: The

text: reﬁnement

text: process

text: can

text: only

text: increase

text: the

text: length

text: of

text: the

text: factorization,

text: so

text: the

text: length

text: of

text: the

text: original

text: factorization

text: is

text: no

text: longer

text: than

text: n,

text: proving

text: the

text: claim.

text: We

text: show

text: for

text: τ-reﬁnable,

text: R

text: a

text: τ-r-UFR,

text: R

text: is

text: a

text: τ-r-atomic

text: τ-r-idf-ring

text: which

text: has

text: been

text: shown

text: in

text: Theorem

text: to

text: be

text: equivalent

text: to

text: being

text: a

text: τ-r-FFR.

text: R

text: being

text: τ-r-factorial

text: gives

text: us

text: τ-r-

text: atomic

text: for

text: free.

text: Furthermore,

text: any

text: τ-atomic

text: factorization

text: of

text: a

text: Reg(R)#

text: has

text: the

text: same

text: length,

text: say

text: n

text: and

text: can

text: be

text: reordered

text: so

text: that

text: the

text: associates

text: match

text: up.

text: This

text: tells

text: us

text: there

text: are

text: precisely

text: n

text: τ-irreducible

text: divisors

text: of

text: a

text: up

text: to

text: associate,

text: hence

text: R

text: is

text: a

text: τ-r-idf-ring.

text: Suppose

text: R

text: is

text: a

text: τ-r-FFR,

text: by

text: deﬁnition,

text: we

text: know

text: R

text: is

text: τ-r-atomic.

text: Now,

text: let

text: a

text: Reg(R)#,

text: let

text: S

text: be

text: the

text: ﬁnite

text: set

text: of

text: all

text: τ-atomic

text: factors

text: of

text: a.

text: Set

text: N(a)

text: S

text: Let

text: a

text: λa

text: a

text: be

text: a

text: n

text: τ-atomic

text: factorization

text: of

text: a,

text: then

text: a

text: S

text: for

text: all

text: i,

text: but

text: then

text: a

text: n

text: S

text: and

text: hence

text: is

text: ﬁnite

text: and

text: i

text: i


text: τ-REGULAR FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS 9 ≤ | | n N(a) = S as desired, so R is a τ-r-BFR. (5) Let R be a τ-r-BFR, and we suppose for a moment that R does not satisfy τ-r-ACCP. { }∞ ⊆ | (cid:54)∼ There must exist and inﬁnite sequence a Reg(R)# such that a a , but a a ≥ · · · i i=1 n+1 τ n ≥n+1 n for all n 1. Let a = λ r r a be a τ factorization of a for all n 1. But n n+1 n+11 n+1sn+1 n+1 n then we have · · · · · · · · · · · · a = λ r r a = λ r r λ r r a = 1 2 21 2s2 2 2 21 2s2 3 31 3s3 3 is a τ factorization (note we use τ reﬁnable here). Furthermore, each of these factorizations can be reﬁned into τ-atomic elements, and it will still be a τ-factorization the length of which ≥ · · · ≥ L (a ) s + s + s + 1 n which shows we can ﬁnd arbitrarily large τ-atomic factoriza- τ 1 2 3 n tions of a which contradicts the fact the R is a τ-r-BFR. 1 (6) Let R satisfy τ-r-ACCP, but suppose that R is not τ-r-atomic. Then there exists ∈ a Reg(R)# with no τ-factorization into τ-atoms. a itself cannot be a τ-atom, so say a = · · · λa a is a τ-factorization with n > 1. Now again some a must not be a product of τ-atoms, 1 n i | (cid:54)∼ or with τ reﬁnable, we could ﬁnd a τ-atomic factorization, say it is a . Then a a and a a · · · 1 1 τ 1 put b = a . Then a must have a τ-factorization a = λ a a where n > 1. Again, one 1 1 1 1 2 21 2n2 | 2 (cid:54)∼ of the τ-factors, say a cannot be a τ-product of τ-atoms. Here a a = b and a a . 21 { }∞21 τ 1 1 21 1 Put b = a . Continuing in this fashion, we obtain a sequence b of elements of Reg(R)# 2 21 | (cid:54)∼ ≥ i i=1 (cid:3) such that b b but b b for every n 1. This contradicts R satisfying τ-r-ACCP. n+1 τ n n+1 n ∇ The following diagram summarizes our result where represents τ being reﬁnable. τ-(cid:50)(cid:58) r-HFR τ-ACCP ∇ (cid:36)(cid:44) (cid:11)(cid:19) ∇ ∇ ∇ (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) (cid:43)(cid:51) τ-r-UFR τ-r-F(cid:75)(cid:83) FR τ-r-BFR τ-r-ACCP τ-r-atomic ∇ (cid:11)(cid:19) τ-r-W(cid:75)(cid:83) FFR ∇ (cid:11)(cid:19) (cid:36)(cid:44) (cid:43)(cid:51) τ-atomic τ-r-idf τ-r-idf 4. τ -Factorizations reg In this section, we study another approach which could have be used to extend τ-factorization to commutative rings with zero-divisors using regular factorizations. In Section , we decided to only consider factorizations of the regular elements. In other words, we chose to restrict the elements we attempt to factor to the regular elements of a commutative ring R. We could have instead chosen to restrict the relation τ itself. This gives us the beneﬁt of not completely ignoring a possible large number of zero-divisors in the ring R, but at the cost of choosing a less natural relation τ. Moreover, it allows us to use much of the work done previously in [17] by just picking a diﬀerent τ and keeping all of the original deﬁnitions the same. It turns out that in many ways, either choice is ﬁne and we end up at the same place anyway. Studying this will be the motivation of this section. Let R be a commutative ring with 1 and τ a symmetric relation on R#. Then we deﬁne a new relation ∩ × τ := τ (Reg(R) Reg(R)) . reg We may now pursue the τ-factorizations using the approach from [17] and look at factoring all the non-units in R instead of just the regular elements. There is certainly a very close ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: τ-REGULAR

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: n

text: N(a)

text: S

text: as

text: desired,

text: so

text: R

text: is

text: a

text: τ-r-BFR.

text: Let

text: R

text: be

text: a

text: τ-r-BFR,

text: and

text: we

text: suppose

text: for

text: a

text: moment

text: that

text: R

text: does

text: not

text: satisfy

text: τ-r-ACCP.

text: There

text: must

text: exist

text: and

text: inﬁnite

text: sequence

text: a

text: Reg(R)#

text: such

text: that

text: a

text: a

text: but

text: a

text: a

text: i

text: τ

text: n

text: n

text: for

text: all

text: n

text: Let

text: a

text: λ

text: r

text: r

text: a

text: be

text: a

text: τ

text: factorization

text: of

text: a

text: for

text: all

text: n

text: But

text: n

text: n

text: then

text: we

text: have

text: a

text: λ

text: r

text: r

text: a

text: λ

text: r

text: r

text: λ

text: r

text: r

text: a

text: is

text: a

text: τ

text: factorization

text: (note

text: we

text: use

text: τ

text: reﬁnable

text: here).

text: Furthermore,

text: each

text: of

text: these

text: factorizations

text: can

text: be

text: reﬁned

text: into

text: τ-atomic

text: elements,

text: and

text: it

text: will

text: still

text: be

text: a

text: τ-factorization

text: the

text: length

text: of

text: which

text: L

text: (a

text: s

text: s

text: s

text: n

text: which

text: shows

text: we

text: can

text: ﬁnd

text: arbitrarily

text: large

text: τ-atomic

text: factoriza-

text: τ

text: n

text: tions

text: of

text: a

text: which

text: contradicts

text: the

text: fact

text: the

text: R

text: is

text: a

text: τ-r-BFR.

text: Let

text: R

text: satisfy

text: τ-r-ACCP,

text: but

text: suppose

text: that

text: R

text: is

text: not

text: τ-r-atomic.

text: Then

text: there

text: exists

text: a

text: Reg(R)#

text: with

text: no

text: τ-factorization

text: into

text: τ-atoms.

text: a

text: itself

text: cannot

text: be

text: a

text: τ-atom,

text: so

text: say

text: a

text: λa

text: a

text: is

text: a

text: τ-factorization

text: with

text: n

text: Now

text: again

text: some

text: a

text: must

text: not

text: be

text: a

text: product

text: of

text: τ-atoms,

text: n

text: i

text: or

text: with

text: τ

text: reﬁnable,

text: we

text: could

text: ﬁnd

text: a

text: τ-atomic

text: factorization,

text: say

text: it

text: is

text: a

text: Then

text: a

text: a

text: and

text: a

text: a

text: τ

text: put

text: b

text: a

text: Then

text: a

text: must

text: have

text: a

text: τ-factorization

text: a

text: λ

text: a

text: a

text: where

text: n

text: Again,

text: one

text: of

text: the

text: τ-factors,

text: say

text: a

text: cannot

text: be

text: a

text: τ-product

text: of

text: τ-atoms.

text: Here

text: a

text: a

text: b

text: and

text: a

text: a

text: τ

text: Put

text: b

text: a

text: Continuing

text: in

text: this

text: fashion,

text: we

text: obtain

text: a

text: sequence

text: b

text: of

text: elements

text: of

text: Reg(R)#

text: i

text: such

text: that

text: b

text: b

text: but

text: b

text: b

text: for

text: every

text: n

text: This

text: contradicts

text: R

text: satisfying

text: τ-r-ACCP.

text: τ

text: n

text: n

text: The

text: following

text: diagram

text: summarizes

text: our

text: result

text: where

text: represents

text: τ

text: being

text: reﬁnable.

text: r-HFR

text: τ-ACCP

text: τ-r-UFR

text: FR

text: τ-r-BFR

text: τ-r-ACCP

text: τ-r-atomic

text: FFR

text: τ-atomic

text: τ-r-idf

text: τ-r-idf

text: τ

text: -Factorizations

text: reg

text: In

text: this

text: section,

text: we

text: study

text: another

text: approach

text: which

text: could

text: have

text: be

text: used

text: to

text: extend

text: τ-factorization

text: to

text: commutative

text: rings

text: with

text: zero-divisors

text: using

text: regular

text: factorizations.

text: In

text: Section

text: we

text: decided

text: to

text: only

text: consider

text: factorizations

text: of

text: the

text: regular

text: elements.

text: In

text: other

text: words,

text: we

text: chose

text: to

text: restrict

text: the

text: elements

text: we

text: attempt

text: to

text: factor

text: to

text: the

text: regular

text: elements

text: of

text: a

text: commutative

text: ring

text: R.

text: We

text: could

text: have

text: instead

text: chosen

text: to

text: restrict

text: the

text: relation

text: τ

text: itself.

text: This

text: gives

text: us

text: the

text: beneﬁt

text: of

text: not

text: completely

text: ignoring

text: a

text: possible

text: large

text: number

text: of

text: zero-divisors

text: in

text: the

text: ring

text: R,

text: but

text: at

text: the

text: cost

text: of

text: choosing

text: a

text: less

text: natural

text: relation

text: τ.

text: Moreover,

text: it

text: allows

text: us

text: to

text: use

text: much

text: of

text: the

text: work

text: done

text: previously

text: in

text: by

text: just

text: picking

text: a

text: diﬀerent

text: τ

text: and

text: keeping

text: all

text: of

text: the

text: original

text: deﬁnitions

text: the

text: same.

text: It

text: turns

text: out

text: that

text: in

text: many

text: ways,

text: either

text: choice

text: is

text: ﬁne

text: and

text: we

text: end

text: up

text: at

text: the

text: same

text: place

text: anyway.

text: Studying

text: this

text: will

text: be

text: the

text: motivation

text: of

text: this

text: section.

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: τ

text: a

text: symmetric

text: relation

text: on

text: R#.

text: Then

text: we

text: deﬁne

text: a

text: new

text: relation

text: τ

text: τ

text: (Reg(R)

text: Reg(R))

text: reg

text: We

text: may

text: now

text: pursue

text: the

text: τ-factorizations

text: using

text: the

text: approach

text: from

text: and

text: look

text: at

text: factoring

text: all

text: the

text: non-units

text: in

text: R

text: instead

text: of

text: just

text: the

text: regular

text: elements.

text: There

text: is

text: certainly

text: a

text: very

text: close

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