IN INTEGRAL DOMAINS
D.D. ANDERSON and J. REINKOESTER
LetDbe an integral domain. We are interested in “relatively prime” relationsτ onD#, the nonzero nonunits ofD, that is, relations that share properties with the three relations: aandbare comaximal,aandbarev-coprime (((a, b)−1)−1=D), andaandbare relatively prime. Of particular interest are relations of the form τS for a set S of proper ideals ofD where a τSb⇔(a, b)*I for each I ∈S.
Examples includeS=?-max(D), the maximal?-ideals of a finite character star operation?onD (for?=d, we get comaximal and for?=t, we getv-coprime), S={(a)|a∈D#}(where we get relatively prime), andS=X(1)(D), the set of height-one prime ideals ofD. We also study τ-factorization of an element a ofD# (a=λa1· · ·an, λa unit, ai ∈D#,aiτ aj fori6=j) into τ-atoms (each τ-factorization has length one).
AMS 2010 Subject Classification: 13G05, 13A05, 13A15.
Key words: relatively prime, star operation, factorization,τ-factorization.
1. INTRODUCTION
Throughout D will be an integral domain with quotient field K and group of units U(D). LetD∗ =D− {0}andD#=D∗−U(D). At least three different notions of “relatively prime” play an important role in commutative algebra. Let a, b∈D#; we say that aand barecomaximal (resp.,v-coprime, relatively prime) if (a, b) =D (resp., ((a, b)−1)−1 =D, [a, b] = 1). Here (a, b) denotes the ideal of D generated by aand band [a, b] denotes the GCD of a and b; we will denote the LCM ofaand b by ]a, b[.
In this paper we are interested in “relatively prime” relations onD#with an eye toward factorization. Recall that a star operation ? on D is a closure operation on F(D), the set of nonzero fractional ideals of D, that satisfies (1) D? = D and (2) (xA)? = xA? for all x ∈ K∗ and A ∈ F(D). For an introduction to star operations, see [7]. Two important star operations are thed-operation,A→Ad=A and thev-operation,A→Av= (A−1)−1, where as usualA−1={x∈K |xA⊆D}. Givena, b∈D#, we say thataand bare
?-comaximal if (a, b)?=Dand write [a, b]? = 1 ora τ? b. Note that for?=d,
REV. ROUMAINE MATH. PURES APPL.,56(2011),2, 85–103
a and b ared-comaximal if and only if aand b are comaximal and for?=v, a andb arev-comaximal if and only ifaand b arev-coprime.
Another example ofa“relatively prime” relation onDis as follows. Let S be a set of ideals ofD. Definea, b∈D# to be S-relatively prime, written [a, b]S= 1 ora τS b, if (a, b)*I for eachI ∈S. If we takeSto be the set of nonzero proper principal ideals ofD, we get [a, b]S= 1 if and only if [a, b] = 1.
We also write a τ[ ] b if [a, b] = 1. For a finite character star operation ? on D, take S? =?-max(D), the set of maximal ?-ideals of D. Then [a, b]S? = 1 if and only if a and bare?-comaximal. We will also be interested in the case where S=X(1)(D), the set of height-one prime ideals of D.
A relationτ on D# is called a generalized relatively prime relation if it satisfies (1) GRP1: a τ b ⇒ [a, b] = 1, (2) GRP2: (a, b) = D ⇒ a τ b, (3) GRP3: a τ b⇒ b τ a, and (4) GRP4: a τ b, a0 |a, b0 |b ⇒a0 τ b0. We also consider the conditions GRP5: a τ b and (a, b) ⊆ (c, d) ⇒ c τ d and GRP6:
a τ b and a τ c ⇒a τ bc. We show that a relation τ on D# satisfies GRP1, GRP2, and GRP5 (and hence GRP3 and GRP4) if and only if τ =τS where S is a set of nonzero proper ideals of D having the property that for each proper principal ideal (a) of D, (a)⊆I for someI ∈S.
Our interest in relatively prime relations onD#comes from factorization.
In [1], the notion of a τ-factorization was introduced. Let τ be a symmetric relation onD#. Then fora∈D#, aτ-factorization of ais a factorizationa= λa1· · ·anwhereλ∈U(D), eachai∈D#, andaiτ aj fori6=j. The relationτ is said to bemultiplicativeif fora, b, c∈D#,a τ banda τ c⇒a τ bcand to be divisiveifa, b, a0, b0 ∈D#witha0|aandb0 |b, thena τ b⇒a0τ b0. An element a∈D# is a τ-atom if adoes not have aτ-factorization a=λa1· · ·an where n > 1. If τ is a divisive relation on D# and D satisfies the ascending chain condition on principal ideals (ACCP), then eacha∈D#has aτ-factorization into τ-atoms (called a τ-atomic factorization) [1]. In Section 3 we consider τ-atomic factorizations for τ a generalized relatively prime relation.
2. RELATIVELY PRIME RELATIONS
We begin with the following proposition listing some well known proper- ties of the comaximal, v-coprime, and relatively prime relations.
Proposition 2.1. Let D be an integral domain and let a, b∈D#. (1) a and b are comaximal if and only if [a, b] = 1 and [a, b] is a linear
combination of aand b.
(2) The following conditions are equivalent:
(a) (a, b)v =D, (b) ]a, b[=ab,
(c) (a)∩(b) = (ab), (d) (a) : (b) = (a),
(e) (b) : (a) = (b),
(f) a, b is an R-sequence (here we drop the condition (a, b) 6= D), and
(g) [ad, bd] =d for alld∈D#.
(3) [a, b] = 1if and only if (a, b)⊆(t)⊆D implies (t) =D.
(4) a, b comaximal ⇒ a, b are v-coprime ⇒ a, bare relatively prime.
Proof. We sketch the proof of (2) and leave the other statements to the reader. The equivalence of (a) and (c) follows from (a, b)−1 = (a−1)∩ (b−1) = ab1((a)∩(b)). The equivalence of (b) and (c) follows from the fact that ]a, b[= d⇔ (a)∩(b) = (d). It is easily checked that (c)−(e) are equivalent and that (f) is equivalent to (d). If (a, b)v =D, then (ad, bd)v = dD which implies [ad, bd] = d, so (a) ⇒ (g). Finally, suppose that [ad, bd] = d for all d∈ D∗. Let (a, b) ⊆ xyD where x, y∈ D∗. Then (ay, by) ⊆xD; so x |ay, by and hence x|[ay, by] =y. So D⊆ xyD. Thus (a, b)v =D, so (g)⇒(a).
We next review some facts about star operations. Given any star opera- tion?onD, we get a new star operation?sonDdefined byA?s =S
{F? |F ∈ F(D), F ⊆A with F finitely generated}. We say that ? has finite character if ?=?s. Thet-operation is given by t=vs. Note that since (a, b) is finitely generated, (a, b)?= (a, b)?s; thus there is no loss in generality for us to assume that ? has finite character; which we shall do. So let ? be a finite character star operation. Then each proper integral ?-ideal (an idealAwithA=A?) is contained in a maximal proper integral?-ideal and a maximal?-ideal is prime.
We denote the set of maximal ?-ideals by?-max(D).
Let ? be a (finite character) star operation on D. We define a, b∈ D# to be ?-comaximal if (a, b)? = D. We denote that a and b are ?-comaximal by [a, b]? = 1 or a τ? b where we are thinking ofτ? as a relation onD#. If we take ? =d, we get the notion of comaximal elements and if ? = t (or v) we have the notion of v-coprime elements.
Now if we replace the star operation ? on D by a closure operation c on the set F+(D) of nonzero integral ideals of D (so Dc =D), we can define a, b∈D# to bec-comaximal if (a, b)c =D. If we let Sc={A∈F+(D)|A= Ac}, then for A ∈ F+(D), Ac =T
{B ∈ Sc |B ⊇ A}. Note that Ac =D if and only if A * I for each I ∈ Sc − {D}. With this in mind we make the following definition.
Definition 2.2. LetSbe a set of ideals ofD. We say thata, b∈D#are S-coprime if (a, b) * I for each I ∈ S. We denote this by [a, b]S = 1. This gives the relation τS on D# defined bya τS b⇔[a, b]S = 1.
We next give some examples.
Examples 2.3. LetDbe an integral domain.
(1) Let S= {D}. Here no two elements a, b ∈ D# are S-coprime and so τS=∅.
(2) Let S={0}. Here any two elements a, b ∈D# are S-coprime and so τS=D#×D#.
(3) Let?be a finite character star operation onD. TakeS=?-max(D).
So for a, b∈D#, [a, b]S= 1⇔(a, b)? *P for anyP ∈?-max(D)⇔(a, b)?= D ⇔ a and b are ?-comaximal. So if we take ? = d, we have a and b are comaximal ⇔[a, b]S= 1 for S= max(D); and for?=t, we haveaand bare v-coprime⇔[a, b]S = 1 for S=t-max(D).
(4) LetS={(t) |t∈ D#}. So a, b∈D# are S-coprime⇔ (a, b) *(t) for any (t)∈S⇔[a, b] = 1.
(5) Fora, b ∈D#, (a, b)v =D ⇔a, b is an R-sequence ⇔G((a, b))>1 (that is, (a, b) contains an R-sequence of length greater than one). There is a corresponding height version. LetS=X(1)(D) be the set of height-one prime ideals ofD. Then [a, b]S = 1⇔(a, b)*P for eachP ∈X(1)(D)⇔D= (a, b) or ht(a, b)>1. We will sometimes denote [a, b]S = 1 by a τX b.
For D Noetherian, the grade version can be made more precise. ForP a nonzero prime ideal of D, Pt 6=D⇔ G(P) = 1. Also, P ∈t-max(D) ⇔P is a maximal prime of a principal ideal, that is, P is a maximal prime ideal contained inZ(D/(a)) for some a∈D#. And of course,P ∈X(1)(D)⇔P is a minimal prime of some a∈D#. We will examine the caseS=X(1)(D) in more detail later.
(6) Let D = k[X, Y], k a field. Take S1 = {M ∈ max(D) | M 6=
(X, Y)} ∪ {P ∈ X(1)(D) | P ⊂ (X, Y)} and S2 = {P ∈ X(1)(D) | P * (X, Y)} ∪ {(X, Y)}. For S1, a τS1 b ⇔ (a, b) =D orp
(a, b) = (X, Y). So, for example, X τS1 Y, but (X+ 1) 6 τS1 Y (even though (X+ 1, Y)v =D).
For S2,a τS2 b⇔D= (a, b) orht(a, b)>1 and (a, b)*M. So, for example, (X+ 1) τS2 Y, butX6τS2 Y (even though (X, Y)v =D).
Definition 2.4. LetDbe an integral domain. A relationτ onD#is called ageneralized relatively prime relation if it satisfies the following conditions for a, b∈D#:
GRP1: a τ b⇒[a, b] = 1 (so a6τ a), GRP2: (a, b) =D⇒a τ b,
GRP3: a τ b⇒b τ a(τ is symmetric), and
GRP4: a τ band a0 |a, b0|b(a0, b0 ∈D#) ⇒a0 τ b0 (τ is divisive).
We will also be interested in generalized relatively prime relations that satisfy either of the following two conditions.
Definition 2.5. Let D be an integral domain and τ a relation on D#. Then τ satisfies GRP5 if for a, b, c, d∈D# with (a, b) ⊆(c, d), thena τ b ⇒
c τ d and τ satisfies GRP6, or is multiplicative, if for a, b, c∈D#,a τ b and a τ c implya τ bc, and b τ aand c τ aimply bc τ a.
For two relationsτ1, τ2 onD#, we writeτ1≤τ2 ifτ1⊆τ2 (as subsets of D#×D#). Note that a relation τ on D# satisfies GRP1-GRP4 if and only if τd ≤ τ ≤ τ[ ] with τ divisive and symmetric. Given two star operations
?1 and ?2 on D, we write ?1 ≤ ?2 if I?1 ⊆ I?2 for each nonzero (integral) ideal of D. Note that ?1 ≤ ?2 if and only if τ?1 ≤ τ?2. So for any (finite character) star operation ? on D we have τd ≤τ? ≤τv (τd ≤τ? ≤τt). Also, τd≤τt≤τX ≤τ[ ]where for the last inequality we need that a proper principal ideal is contained in a height-one prime ideal. Note that for a nonempty setS of proper ideals of an integral domain D,τS trivially satisfies GRP2, GRP3, GRP4, and GRP5. AndτS satisfies GRP1 if and only if each proper principal ideal of D is contained in some element of S. So Example 2.3 (1) does not satisfy GRP1 and τX of Example 2.3 (5) will satisfy GRP1 if and only if for each a∈D#, ht(a) = 1; which of course holds ifD is Noetherian. Note that GRP5 ⇒ GRP3, GRP4. Examples 2.3 (2)–(6) (with DNoetherian for τX in (5)) satisfy GRP1–GRP5. We next consider GRP6.
Proposition 2.6. Let D be an integral domain and let S be a set of prime ideals of D. Then τS satisfies GRP6, that is, τS is multiplicative.
Proof. Leta, b, c∈D#and suppose that a τS b and a τS c. LetP ∈S.
Then (a, b) * P and (a, c) * P. Since P is prime, (a, b)(a, c) * P. But as (a, b)(a, c)⊆(a, bc); we must have (a, bc)*P. So a τS bc.
Example 2.3 (1) satisfies GRP6 vacuously and Examples 2.3 (2), (3), (5) and (6) satisfy GRP6 by Proposition 2.6. In the case of Example 2.3 (3) where
? is a star operation on D, the fact thatτ? satisfies GRP6 also follows from:
if (a, b)? =D and (c, d)? =D, then ((a, b)(c, d))? = ((a, b)?(c, d)?)? =D.
However, in generalτ[ ]in Example 2.3 (4) need not satisfy GRP6, that is, [a, b] = [a, c] = 1 need not imply that [a, bc] = 1. For example, let (D, M) be a one-dimensional local domain that is not a DVR. Leta, b∈Dbe nonassociate irreducible elements. Now p
(a) = M =p
(b), so there exists a least n > 1 with (bn) ⊆ (a). Then [a, bn−1] = 1, but [a,(bn−1)2] 6= 1 (when we write [c, d]6= 1, we mean thatcand dare not relatively prime; we are not assuming that [c, d] exists). A domainD satisfies Gauss’ Lemma if the product of two primitive polynomials of D[X] is again primitive. It can be shown that D satisfies Gauss’ Lemma if and only if τ[ ] satisfies GRP6 [4]. It is well known that in a domain satisfying Gauss’ Lemma, atoms are prime. Thus for an atomic domain D,τ[ ] satisfies GRP6 if and only ifDis a UFD. Suppose that D is an integral domain andS={(t)|t∈D#}; soτS =τ[ ]. IfD is atomic, we can replace Sby Sa={(a)|a∈D is an atom}. Thenτ[ ]=τSa satisfies GRP6 if and only ifSa⊆Spec(D).
We next characterize when a relationτ on D# has the form τ =τS for some set Sof ideals of D.
Theorem 2.7. Let D be an integral domain.
(1) A relation τ on D# is given by a set S of ideals, that is, τ =τS, if and only if τ satisfiesGRP5.
(2)A relation τ onD# is given by a setS consisting of proper ideals if and only if τ satisfiesGRP2 and GRP5.
(3) A relation τ on D# is given by a set of proper ideals S having the property that for eacha∈D#,(a)⊆I for someI ∈Sif and only ifτ satisfies GRP1, GRP2, and GRP5 (and hence GRP3 andGRP4).
Proof. (1) (⇒) Clear. (⇐) Suppose that τ satisfies GRP5. Let S = {(a, b) | a 6 τ b}. If S = ∅, a τ b for all a, b ∈ D#. As in Example 2.3 (2), we can take S = {0}. So suppose S 6= ∅. Suppose that a τ b. We need [a, b]S = 1, that is, (c, d) ∈ S where c 6 τ d ⇒ (a, b) * (c, d). But if (a, b)⊆(c, d), then a τ b and GRP5 givesc τ d, a contradiction. Conversely, suppose that [a, b]S= 1. So (a, b)*(c, d) ifc6τ d. Suppose thata6τ b. Then (a, b) ⊆(a, b) ∈ S, a contradiction. (2) and (3) follow immediately from (1) and previous comments.
We next give an example of a generalized relatively prime relation τ that does not satisfy GRP5 and hence does not have the form τ =τS. Thus GRP1-GRP4 6⇒ GRP5. We then give an example of a relation τS satisfying GRP1-GRP6 that is not given by a star operation.
Example 2.8. GRP1-GRP4 6⇒ GRP5. Take D = k[X, Y], k a field.
For f, g ∈ D#, define f τ g ⇔ (1) (f, g) = D, (2) f = αX, g = βY, or (3) f =αY, g=βX (α, β ∈k∗). Clearly,τ satisfies GRP1-GRP4. However, X τ Y and (X, Y)⊆(X+Y, Y), but (X+Y)6τ Y.
Example 2.9. τ satisfies GRP1-GRP66⇒ τ =τ? for some star operation
?. Let (D, M) be a two-dimensional local domain with G(M) = 1, that is,D is not Cohen-Macaulay. Let S= X(1)(D), so τS satisfies GRP1-GRP6. Let
? be a star operation on D. NowM ⊆M? ⊆Mt=M; so ?-max(D) ={M}.
Now for a, b ∈ M − {0}, (a, b)? 6= D, so a 6 τ? b. But if a, b is a system of parameters for D, p
(a, b) = M; so (a, b) * P for each P ∈ X(1)(D). So a τS b. ThusτS 6=τ?. In fact, it is easily seen that for a Noetherian domain D and S=X(1)(D), the following are equivalent: (1) τS = τ? for some star operation ?, (2) τS =τt, (3) each grade-one prime ideal ofD has height one, (4) every maximal t-ideal ofDhas height one.
We have shown that ifSconsists of prime ideals, thenτS is multiplica- tive. Of course,τS can be multiplicative withoutSconsisting of prime ideals.
For example, we can always add non-prime ideals to max(D) to get a setSnot
consisting of prime ideals, but withτS multiplicative. So the correct question to ask is the following: If S is a set of proper ideals of an integral domain with τS multiplicative and with the property that for eacha ∈D#, (a) ⊆ I for some I ∈ S, does there exist a subset S0 ⊆ Spec(D) so that τS0 = τS? While we do not know the answer to this question, we do have the following related result.
Theorem2.10. Let Dbe an integral domain andSa set of ideals ofD.
Then the following conditions are equivalent.
(1)τS=τS0 for some collection S0 of radical ideals.
(2)For a, b∈D#, a τS b⇒an τS bm for all n, m≥1.
(3)For a, b∈D#, a τS b⇒a τS b2.
Proof. Since (1),(2), and (3) hold if S = ∅, we assume that S 6= ∅.
(1) ⇒ (2) Since τS = τS0, we can assume that S consists of radical ideals.
Suppose that a τS b. Then for I ∈ S, (a, b) * I. Since I is a radical ideal (an, bm) * I, so an τS bm. (2) ⇒ (3) Clear. (3) ⇒ (1) Take S0 = {√
I | I ∈ S}. We show that τS = τS0. Suppose a τS0 b. So if I ∈ S, √
I ∈ S0; thus (a, b) *
√
I. Hence (a, b) * I; so a τS b. Next suppose that a τS b.
Let I ∈ S; so (a, b) * I. We need (a, b) *
√
I. Suppose that (a, b) ⊆ √ I. Then there exist n, m ≥ 1 with an, bm ∈ I, so (an0, bm0) ⊆ I for n0 ≥ n and m0 ≥ m. But a τS b ⇒ a τS b2 ⇒ a τS b2k for all k ≥ 0. Also, a τS b2k ⇒ b2k τS a ⇒ b2k τS a2l for all l ≥ 0. Hence, a2l τS b2k for all l, k ≥0; i.e., (a2l, b2l)*I. But this is a contradiction.
3. RELATIVELY PRIME FACTORIZATION
In this section we study factorization into “relatively prime” elements.
We first review the general theory of factorization involving τ-factorizations introduced in [1].
Throughout this section D will be an integral domain and τ a sym- metric relation on D#. For a ∈ D#, a factorization a = λa1· · ·an where λ ∈ U(D), ai ∈ D#, n≥ 1 and ai τ aj fori 6= j is called a τ-factorization of a. In this case we sayai τ-divides a, written ai |τ a. We calla=λ(λ−1a) a trivial τ-factorization of a and a is a τ-atom (or τ-irreducible) if the only τ-factorizations of a are trivial. A τ-factorization a = λa1· · ·an is τ-atomic if each ai is a τ-atom and D is τ-atomic if each a∈D# has a τ-atomic fac- torization. D is a τ-UFD if (1) D is a τ-atomic and (2) given two τ-atomic factorizations λa1· · ·an = µb1· · ·bm, then n = m and after re-ordering, if necessary, ai∼bi fori= 1, . . . , n.
Letτ1and τ2 be symmetric relations onD#,Dan integral domain, such that τ1 ≤ τ2. Then a τ1-factorization is a τ2-factorization. Hence a τ2-atom is a τ1-atom. Since τd ≤ τt ≤ τX ≤ τ[ ] (where for the last inequality we assume that each a∈ D# is contained in a height-one prime ideal), we have the following implications: prime⇒atom⇒ τ[ ]-atom⇒τX-atom⇒τt-atom
⇒ τd-atom.
An element a ∈ D# is said to be τ-prime (resp., |τ-prime) if for each τ-factorization λa1· · ·an, a | λa1· · ·an (resp., a |τ λa1· · ·an) implies a |ai (resp., a|τ ai) for somei. So aτ-prime is a τ-atom. Note that our definition of τ-atom, τ-prime, and |τ-prime involve τ-factorizations of arbitrary length;
not just those of length two (which is the case in the usual definition that ais an atom (resp., prime) if a6=bc (resp., a|bc⇒ a|b ora|c), a, b, c ∈D#).
This is essential. For example, if K ( L are fields and D = K+XL[[X]], then D is τ[ ]-atomic, but D has an element that has a τ[ ]-factorization of length three (and hence is not a τ[ ]-atom) with noτ[ ]-factorization of length two [1, Section 4]. However, this can not happen if τ is multiplicative. For if λa1· · ·an is aτ-factorization with τ multiplicative, then so isλb1· · ·bs where {1,2, . . . , n}=A1
∪ · · ·· ∪· As (disjoint union) and bi=Q
{aj |j∈Ai}.
If τ is divisive (or more generally, associate preserving: a τ b and b ∼ c ⇒ a τ c), then a = λa1· · ·an is a τ-factorization if and only if a = a1· · ·(λai)· · ·an is a τ-factorization. Hence, in this case we can dispense with the unit λ, and we shall do so. From now on we shall assume that τ is divisive. If a =a1· · ·ai· · ·an and ai = b1· · ·bm are τ-factorizations, then a = a1· · ·ai−1b1· · ·bmai+1· · ·an is a τ-factorization. A τ-factorization of a obtained by τ-factoring one or more of the a0is is called a τ-refinement of a.
A τ-factorization with no proper τ-refinements is said to be τ-complete and D is τ-complete if eacha∈D# has a τ-complete factorization. Clearly, a τ- atomic factorization (resp.,τ-atomic domain) isτ-complete and for τ divisive the converse is true.
We have the following result from [1].
Theorem3.1. Let Dbe an integral domain andτ a symmetric and divi- sive relation on D# (e.g., τ is a generalized relatively prime relation on D#).
(1) If D satisfies ACCP, then D is τ-atomic[1, Theorem 2.9].
(2) If D is a UFD, D is a τ-UFD [1,Theorem 2.11].
We first consider domainsDin which every nonzero nonunit is aτ-atom.
Note that fields trivially have this property. Such domains are of course τ- UFD’s.
Theorem 3.2. Let D be an integral domain.
(1)Every element of D# is a τd-atom ⇔ D is quasilocal.
(2) Every element of D# is a τt-atom ⇔ for a, b ∈D#, (a, b)t 6=D ⇔ (D, M) is quasilocal and for a, b∈M− {0}, (a, b)t⊆M.
(3) Every element of D# is a τX-atom ⇔ for a, b ∈ D#, (a, b) is con- tained in a height-one prime ideal ⇔ (D, M) is quasilocal and for a, b ∈ M − {0}, ht(a, b) = 1.
(4) Every element of D# is a τ[ ]-atom ⇔ for a, b∈D#, there exists a c∈D#with(a, b)⊆(c) ⇔(D, M) is quasilocal and fora, b∈M, there exists a c∈M with(a, b)⊆(c). In the last two equivalences we can replace (a, b) by a finitely generated ideal.
Proof. (1) (⇐) Clear. (⇒) Suppose thatD has more than one maximal ideal, say M1 6=M2 are maximal ideals. Then M1+M2 =D; so there exists mi∈Mi with m1+m2 = 1. Som1m2 is not a τd-atom.
(2) Suppose that every element ofD# is aτt-atom. Thus every element ofD#is aτd-atom; soDis quasilocal by (1). Ifa, b∈D#, thenabis aτt-atom so (a, b)t6=D; or equivalently,a, b∈M− {0}implies (a, b)t⊆M. The reverse implications are obvious.
(3), (4) The proofs of (3) and (4) are similar to the proof of (2).
Corollary 3.3. (1)Suppose that D is a Noetherian domain that is not a field. Then every nonzero nonunit ofD is aτt-atom(resp.,τX-atom) if and only if (D, M) is local and G(M) = 1 (resp., ht(M) = 1).
(2) Suppose that D is an atomic domain that is not a field. Then every nonzero nonunit is a τ[ ]-atom if and only if D is a DVR.
Proof. (1) The τt case follows from remarks made in Example 2.3 (5).
The τX case follows from the converse of the Principal Ideal Theorem.
(2) (⇐) Clear. (⇒) Suppose that Dis atomic with nonassociate atoms a1 anda2. Then there is a c∈D# with (a1, a2)⊆(c), a contradiction.
Gilmer and Heinzer [8] considered condition (K) on a commutative ring R: every proper ideal (a, b) ofR is contained in a proper principal ideal (c).
So the domains in Theorem 3.2 (4) are just the quasilocal domains satisfying condition (K). Such domains remain to be characterized. Also, regarding the domains in Theorem 3.2 (2), it appears to be unknown whether a quasilocal domain (D, M) with (a, b)t⊆M for each a, b∈M− {0}must have Mt6=D.
We next investigateτ[ ]-UFD’s. By Theorem 3.1 (2), a UFD is aτ[ ]-UFD.
But this is easy to see directly. For a UFD D, the τ[ ]-atoms have the form λpn where λ∈U(D), p is a principal prime andn≥1. A τ[ ]-factorization in Disλpn11· · ·pnss whereλ∈U(D),p1, . . . , ps are nonassociate principal primes and each ni ≥ 1. Thus D is τ[ ]-atomic and any two τ[ ]-factorizations are unique up to order and associates. We first give an example of a τ[ ]-UFD that is not a UFD.
Theorem 3.4. Let k ⊆ K be fields, n a positive integer and D = k+ XnK[[X]]. ThenDisτ[ ]-atomic, butDis aτ[ ]-UFD if and only if either(1) k=K and n= 1 (so D=k[[X]] is a DVR), or (2) k=GF(2), K=GF(22) and n= 1 (so D=GF(2) +XGF(22)[[X]]).
Proof. Note thatDis a bounded factorization domain and hence satisfies ACCP. So by Theorem 3.1 (1), Disτ[ ]-atomic.
(⇒) Suppose thatn >1. ThenXn, (1+X)Xn, Xn+1, and (1+X)Xn+1 are nonassociate atoms of Dand thusXn·(1 +X)Xn+1 = (1 +X)Xn·Xn+1 are two different τ[ ]-atomic factorizations of X2n+1 +X2n+2. So D a τ[ ]- UFD implies n = 1. So assume that n = 1. Suppose that |K∗/k∗|> 3. So there exist u, v ∈ K∗ with uv 6= u, v,1 in K∗/k∗. Then X, uX, vX, uvX are nonassociate atoms in D. So uX ·vX = X·uvX are two nonassociate τ[ ]- atomic factorizations inD. Next suppose that|K∗/k∗| ≤3. Then by Brandis’
Theorem k=K orK is finite. So assumek(K. Hence|K∗|=pnm−1 and
|k∗|= pm−1 for some primep and m, n ≥ 1. Now 3≥ |K∗/k∗|= ppnm−1m−1 = (pm)n−1+· · ·+pm+ 1 implies p= 2, m= 1 andn= 2.
(⇐) D=k[[X]] is a UFD and hence aτ[ ]-UFD. Consider D=GF(2) + XGF(22)[[X]]. LetGF(22)∗ =hδi={1, δ, δ2}. Forb=b0+b1X+b2X2+· · · ∈ U(GF(22)[[X]]), b = b0(1 +b−10 b1X +b−10 b2X2 +· · ·) ∈ b0U(D). So the atoms of D are λX, λδX, λδ2X where λ ∈ U(D). Now for aXn, bXm ∈ D where a, b ∈ U(D), [aXn, bXm] = 1 ⇔ n = m = 1 and aU(D) 6= bU(D) in U(D)/U(D) ∼= GF(22)∗/GF(2)∗ = GF(22)∗ = hδi. So the only τ[ ]-atomic factorizations of length greater than one in Dup to order and units of Dare δX2 =X·δX, δ2X2=X·δ2X, andX2 =δX·δ2X. SoDis aτ[ ]-UFD.
We made the following definition.
Definition 3.5. An integral domain Dis square-free factorial (SF-facto- rial) if (1)Dis atomic and (2) if{a1, . . . , an}and{b1, . . . , bm}are two sets of nonassociate atoms witha1· · ·an=b1· · ·bm, thenn=mand after reordering, if necessary, ai∼bi.
Certainly a UFD is SF-factorial. Suppose that D is an atomic domain that is a τ[ ]-UFD. Then D is SF-factorial. This follows since a factorization into nonassociate atoms is a τ[ ]-atomic factorization. Note that GF(2) + XGF(22)[[X]] is an atomic τ[ ]-UFD and hence is SF-factorial, but is not factorial.
It seems appropriate to first investigate when a GCD domain is a τ[ ]- UFD. Note that for a GCD domain D and a, b ∈ D#, (a, b)t = (c) where c = [a, b]. So here τ[ ]=τt. Two obvious examples of GCD domains that are τ[ ]-UFD’s are UFD’s and valuation domains. Recall that a nonzero nonunit element a of a domain D is said to be primal if whenever a | b1· · ·bn, we
can write a= a1· · ·an where ai |bi for each i, and D is pre-Schreier if each a ∈ D# is primal. Certainly a GCD domain is pre-Schreier. Note that for a Schreier domain D, τ[ ] is multiplicative (and of course divisive). For if [a, b] = [a, c] = 1 and x|a and x|bc, then x =x1x2 where x1 |b, x2 |c. So x1|aandx1|b⇒x1 is a unit and likewisex2 is a unit; so [a, bc] = 1.
Theorem 3.6. Let D be a pre-Schreier domain and let a∈D#.
(1)The following are equivalent: (a) is aτ[ ]-atom, (b) ais aτ[ ]-prime, (c) a is a|τ[ ]-prime.
(2)If ahas a τ[ ]-atomic factorization, then it is unique up to order and associates.
(3)D is a τ[ ]-UFD if and only if D isτ[ ]-atomic.
Proof. (1) (a) ⇒ (b) Suppose a is a τ[ ]-atom and a | a1· · ·an where [ai, aj] = 1. Then a = a01· · ·a0n where a0i | ai. Now [ai, aj] = 1 implies [a0i, a0j] = 1. Since a is a τ[ ]-atom, exactly one a0i is a nonunit; so a∼a0i and hencea|ai. (b)⇒(c) This is true for any relationτ that is both multiplicative and divisive [1, Proposition 2.4 (2)]. (c)⇒(a) This is always true.
(2) Suppose thatp1· · ·pn=q1· · ·qm are twoτ[ ]-atomic factorizations of a. Now by (1) each pi, qj is a τ[ ]-prime. So p1 |qi for some iand qi |pk for somek. Thusp1 |pk, so [pi, pj] = 1 fori6=j givesk= 1. Sop1 ∼qi. We can cancel p1 from both sides and proceed by induction to get n=m and pi ∼qi for each iafter re-ordering.
(3) This is clear from (2).
Theorem 3.7. Let D be a GCD domain and {Xα} a set of indetermi- nates over D. Then D is a τ[ ]-UFD if and only if D[{Xα}]is a τ[ ]-UFD.
Proof. (⇐) Clear. (⇒) Suppose D is a τ[ ]-UFD. Since a polynomial involves only finitely many indeterminates, by induction it suffices to show that D is a τ[ ]-UFD implies D[X] is a τ[ ]-UFD. And by Theorem 3.6 it is enough to show that D[X] is τ[ ]-atomic. Let f ∈ D[X]#; so f = ag where a ∈ D∗ and g is primitive. Now either a is a unit or a = a1· · ·an is a τ[ ]- atomic factorization inD(and hence inD[X]). If degg≥1,gis a product of primitive irreducible polynomials each of which is actually prime. Combining these associate primes gives a τ[ ]-atomic factorization forf.
Let D be a GCD domain and let Nt = {f ∈ D[X] | c(f)t = D}. The Kronecker function ring forDwith respect to thet-operation (orv-operation) is Dt ={fg |f, g∈D[X]∗ |c(f)t⊆c(g)t} ∪ {0}=D[X]Nt. Recall thatDt is a Bezout domain and that ADt∩K = At for any nonzero finitely generated ideal Aof D.
Theorem 3.8. Let D be a GCD domain. Then D is a τ[ ]-UFD if and only if Dt is a τd-UFD.
Proof. First observe that for a, b ∈ D#, a τ[ ] b in D ⇔ [a, b] = 1 ⇔ (a, b)t=D⇔(a, b)Dt=Dt⇔a τdbinDt. Thusa1· · ·anis aτ[ ]-factorization inD⇔a1· · ·anis aτd-factorization inDt. Also,a∈D#is aτ[ ]-atom inD⇔ ais a τd-atom inDt. Here (⇐) follows from the previous statement. Suppose a has a nontrivial τd-factorization in Dt, say a=F G where F, G ∈ Dt with (F, G) =Dt. NowF =bfg and G=chk whereb, c∈D∗ andf, g, h, k∈Nt. So a=bfgchk gives agk=bcf h and hence (a) =c(agk)t=c(bcf h)t= (bc). Also, Dt = (F, G) = (b, c) sob τ[ ]c inD. Thus ahas a nontrivial τ[ ]-factorization in D. Hence a has a τ[ ]-atomic factorization in D if and only if a has a τd- atomic factorization in Dt; so D is τ[ ]-atomic if and only if Dt is τd-atomic.
By Theorem 3.6, D is aτ[ ]-UFD if and only ifDt is aτd-UFD.
Recall that an integral domain D has finite character (resp., t-finite character) if the intersection D = T
M∈max(D)
DM (resp., D = T
P∈t- max(D)
DP) is locally finite, that is, each x ∈ D# is in only finitely many M ∈ max(D) (resp., P ∈t-max(D)). An integral domain D has t-dimension one, denoted t-dimD = 1, if every maximal t-ideal of D has height one, or equivalently, t-max(D) = X(1)(D). (Hence if t-dimD = 1, then D = T
P∈X(1)(D)
DP.) An integral domain D is weakly factorial [2] if each nonzero nonunit of D is a product of primary elements and D is a generalized Krull domain if D =
T
P∈X(1)(D)
DP is locally finite and each DP is a valuation domain. Now D is weakly factorial if and only ifD= T
P∈X(1)(D)
DP is locally finite andClt(D) = 0 whereClt(D) is thet-class group ofD[5, Theorem], and for an integral domain D the following are equivalent: (1) D is a weakly factorial GCD domain, (2) D is a weakly factorial generalized Krull domain, (3)Dis a generalized Krull domain and a GCD domain [2, Theorem 20]. Thus to this list we can add the equivalence: D is at-finite character GCD domain witht-dimD= 1.
Theorem 3.9. Let D be a GCD domain.
(1)If D has t-finite character, thenD is a τ[ ]-UFD.
(2) If t-dimD = 1 and D is a τ[ ]-UFD, then D has t-finite character and hence is a generalized Krull domain.
Proof. (1) If Dhas t-finite character, thenDt has finite character. But a finite character Bezout domain is a τd-UFD [10, Corollary 1.10]. But then by Theorem 3.8, D is aτ[ ]-UFD.
(2) Supposet-dimD= 1 andD is aτ[ ]-UFD. Then by Theorem 3.8, Dt is a τd-UFD and dimDt= 1. But aτd-UFD of Krull dimension one has finite character [1, Theorem 4.9]. Hence Dhast-finite character.
We next generalize (at least in the case whereτ is also multiplicative) the result (Theorem 3.1 (2)) that ifD is a UFD andτ is a divisive relation on D, then Dis aτ-UFD. In Theorem 3.12 we show that if Dis a τ[ ]-atomic GCD domain (e.g., a UFD) and τ is a divisive, multiplicative relation on D, then D is a τ-UFD ifD is τ-atomic (which is always the case ifD is a UFD since a UFD satisfies ACCP and hence by Theorem 3.1 (1) is τ-atomic). We need the following two lemmas, the first of which is taken from [1, Lemma 2.10].
Lemma3.10. LetDbe an integral domain and letτ be a divisive relation on D#. Let a1· · ·an be a τ-atomic factorization. Then for i 6= j, either [ai, aj] = 1 or ai ∼aj are atoms.
Lemma 3.11. Let D be a τ-atomic GCD domain where τ is a divisive relation on D. Then the τ[ ]-atoms of D are τ-atoms or elements of the form upn where u is a unit, n≥1, andp is prime.
Proof. Letx be a τ[ ]-atom inD. SinceD isτ-atomic, x has a τ-atomic factorization x=a1· · ·an. If [ai, aj] = 1 for somei6=j, then by Lemma 3.10 we get a proper τ[ ]-factorization of x by grouping the elements that are not relatively prime together. Hence, if n >1, then by Lemma 3.10 the ai must be associative atoms. Moreover, sinceDis a GCD domain, eachai is actually prime. So, x=upn as desired.
Theorem 3.12. Suppose that D is a τ[ ]-atomic (or equivalently, a τ[ ]- UFD) GCD domain and τ is a divisive, multiplicative relation on D. If D is τ-atomic, then D is aτ-UFD.
Proof. By Theorem 3.6Dis aτ[ ]-UFD. We will use this fact along with Lemmas 3.10 and 3.11 throughout this proof without further comment.
We have only to show the uniqueness ofτ-atomic factorizations. Suppose that b1· · ·bm =c1· · ·cn are twoτ-atomic factorizations. If m = 1 orn = 1, then we are done. If [bi, bj] 6= 1 for some i 6= j, then we can group all such τ-atoms, and after grouping and reordering we can write b1· · ·bm = b1· · ·bm0pm1 1· · ·pm00mm00 where eachpi is prime,pi τ pi, and any two factors on the right are relatively prime. We can group theci’s in a similar manner to get
b1· · ·bm0p1m1· · ·psms =c1· · ·cn0q1n1· · ·qtnt.
Note thatbiqjnj for anyj and 1≤i≤m0 sinceqj τ qj. Similarly,cipjmj
for any j and 1≤i≤n0.
Now this element has aτ[ ]-atomic factorization, say
(1) b1· · ·bm0p1m1· · ·psms =a1· · ·ak =c1· · ·cn0q1n1· · ·qtnt.
Also, any pair of elements in the factorization on the left are relatively prime.
Likewise, for the factorization on the right. Hence, since Dis a τ[ ]-UFD, any bi or ci in Equation (1) is a product of a subset of the ai’s, and each pimi or qiniis equal to someai. So bothb1· · ·bm0p1m1· · ·psms andc1· · ·cn0q1n1· · ·qtnt
have τ[ ]-factorizations of the form
(2) (a1,1· · ·a1,s1)(a2,1· · ·a2,s2)· · ·(av,1· · ·av,sv), where, for example,
b1 = (a1,1· · ·a1,s1), b2 = (a2,1· · ·a2,s2), . . . , psms = (av,1· · ·av,sv) (in this instancesv = 1). Let us assume that Equation (2) is a factorization of b1· · ·bm0p1m1· · ·psms. Ifc1· · ·cn0q1n1· · ·qtnt has the same such factorization, then m = n and after reordering bi ∼ ci. We claim that they both must have the same such factorization of the ai’s. If the grouping of factors differs for c1· · ·cn0q1n1· · ·qtnt, then there exists ai,j that is no longer in the same grouping of factors. If ai,j is with a new grouping of factors, then using multiplicativity and the fact that ai,j τ as,t for i6=s we would have a proper τ-factorization of one of the ci’s, a contradiction. If ai,j is not with a new grouping of factors, then ai,j is in a grouping that is a subset of the grouping ai,j was in forb1· · ·bm0p1m1· · ·psms. But thenbihas a proper τ-factorization, a contradiction.
Let D be an integral domain. By abuse of language we say that q is P-primary if (q) is P-primary. It is easily seen that (1) if q1 and q2 are P- primary, then q1q2 is P-primary and (2) if q is P-primary and t | q where t ∈ D#, then t is P-primary. Also, if q is P-primary (with q 6= 0), then P is a maximal t-ideal and hence is the only maximal t-ideal containing (q).
For suppose Q 6= P is a maximal t-ideal containing (q) and let x ∈ Q−P. Now since x /∈ P, (q) : (x) = (q). Hence by Proposition 2.1, (q, x)v = D, a contradiction. Thus if q1 is P1-primary and q2 is P2-primary with P1 6= P2
(and both nonzero), (q1, q2)v = D and hence [q1, q2] = 1. Suppose that D is weakly factorial. Let x ∈ D#; so x is a product of primary elements. By combining elements with the same radical, we can write x = q1· · ·qn where qi is Pi-primary and Pi 6= Pj for i 6= j. Since (x) = (q1)∩ · · · ∩(qn) is a reduced primary decomposition for (x), P1, . . . , Pn are uniquely determined.
Moreover qi is unique up to associates as (qi) = (x)Pi ∩D. Also, each Pi is a height-one maximal t-ideal. Since (qi, qj)v =D for i6= j and a factor of a primary element is primary, x=q1· · ·qn is aτt-atomic factorization forx. By the previous remarks, Dis aτt-UFD. Now while aτ[ ]-atom must be primary, the qi’s need not be τ[ ]-atoms. In fact, in a rank-one nondiscrete valuation
domain V, every element of V# is a τt-atom, but no element of V# is a τ[ ]- atom. We next investigate when a weakly factorial domain is τ[ ]-atomic or a τ[ ]-UFD.
Theorem 3.13. Let Dbe a weakly factorial integral domain.
(1) If a∈ D# is a τ[ ]-atom, then a is P-primary for some height-one prime ideal P of D.
(2) Let 0 6= q ∈ D# be a P-primary (so P ∈ X(1)(D)). Then q is a τ[ ]-atom in Dif and only if q is a τ[ ]-atom in DP.
(3) D is τ[ ]-atomic if and only if DP is τ[ ]-atomic for each height-one prime ideal P of D.
(4) D is a τ[ ]-UFD if and only if DP is a τ[ ]-UFD for each height-one prime ideal P of D.
Proof. (1) We discussed this in the previous paragraph.
(2) Suppose that 06=q is P-primary. (⇒) Suppose that q is aτ[ ]-atom in D; but not in DP. So q = qt1
1 · · ·qtn
n in DP where qi is P-primary, ti ∈/ P, and [qti
i,qtj
j] = 1 in DP fori6=j. Nowt1· · ·tnq=q1· · ·qn isP-primary. Since each nonunit factor of t1· · ·tnq is P-primary and ti ∈/ P, each ti is a unit in D. So with a change of notation, q = q1· · ·qn where qi ∈ D# is P-primary and [qi, qj] = 1 in DP. But then [qi, qj] = 1 in D. For if (qi),(qj) ⊆(t) 6=D, then (t) is P-primary, and hence (qi)P,(qj)P ⊆(t)P 6=DP. So [qi, qj]6= 1 in DP, a contradiction. (⇐) Suppose that q is aτ[ ]-atom inDP, but not in D.
So q = q1· · ·qn where qi ∈ D# with [qi, qj] = 1 for i 6= j. Then each qi is P-primary. Now q =q1· · ·qn in DP#. Assume that some [qi, qj]6= 1 in DP for i6= j. So (qi)P,(qj)P ⊆(t)P where we can take t to be P-primary in D.
But then (qi),(qj)⊆(t); so [qi, qj]6= 1, a contradiction.
(3) (⇒) Suppose thatDisτ[ ]-atomic. Letx∈DP#whereP ∈X(1)(D).
So x = qs where q is P-primary and s /∈ P. It suffices to show that q has a τ[ ]-atomic factorization inDP. Letq=q1· · ·qn be aτ[ ]-atomic factorization ofq inD. SinceqisP-primary, so is eachqi. Then by (2) and its proof eachqi
is a τ[ ]-atom in DP and [qi, qj] = 1 fori6=j inDP. (⇐) Suppose that DP is τ[ ]-atomic for eachP ∈X(1)(D). Letx∈D#. Sox=q1· · ·qn whereqi isPi- primary with Pi 6=Pj fori6=j (Pi ∈X(1)(D)). Since eachDPi is τ[ ]-atomic, we can write qi =qi,1· · ·qi,ni where qi,j is Pi-primary and a τ[ ]-atom in DPi, and [qi,l1, qi,l2] = 1 forl1 6=l2 inDPi. Then q=Qn
i=1
Qnl
l=1qi,l where eachqi,j
is Pi-primary and a τ[ ]-atom inD. Moreover, [qi,l1, qi,l2] = 1 for l1 6=l2 and [qi1,l1, qi2,l2] = 1 forl16=l2 since [qi1, qi2] = 1.
(4) We already know that D is τ[ ]-atomic if and only if DP is τ[ ]- atomic for each P ∈ X(1)(D). (⇒) Suppose that D is a τ[ ]-UFD. If some DP, P ∈ X(1)(D), is not a τ[ ]-UFD, then some P-primary element q of D
has two nonassociate τ[ ]-atomic factorizations in DP. But this gives rise to two nonassociate τ[ ]-atomic factorizations for q in D, a contradiction. (⇐) Suppose that each DP is aτ[ ]-UFD. Suppose that Dis not aτ[ ]-UFD. Since each x ∈ D# has a unique factorization (up to order and associates) in the form x=q1· · ·qnwhereqi isPi-primary withPi 6=Pj fori6=j (Pi∈X(1)(D) and since aτ[ ]-atom is primary,Dnot a τ[ ]-UFD gives that someP-primary elementq ∈D#has two nonassociateτ[ ]-atomic factorizations inD. But this gives two nonassociate τ[ ]-atomic factorizations of q in DP; a contradiction since DP is aτ[ ]-UFD.
Recall that an atomic integral domain D with only a finite number of nonassociate atoms is called aCohen-Kaplansky(CK)domainafter I.S. Cohen and I. Kaplansky who first studied them in [6]. Also, see [3] for a thorough study of CK domains. Now a CK domain is one-dimensional semilocal. And a semiquasilocal domain (D, M1, . . . , Mn) is a CK domain if and only if each DMi is a CK domain. Moreover, a CK domain being Noetherian isτ[ ]-atomic and since a CK domain is weakly factorial, by Theorem 3.13, Dis a τ[ ]-UFD if and only if each DMi is a τ[ ]-UFD. We are interested in when a local CK domain is a τ[ ]-UFD. Of course, a DVR is a CK domain that is a τ[ ]-UFD and by Theorem 3.4,GF(2) +XGF(22)[[X]] is aτ[ ]-UFD with exactly three nonassociate atoms.
LetDbe an atomic integral domain. A subsetS ofDisuniversal if each s∈ S is divisible by each atom of D. Suppose that (D, M) is a CK domain.
Cohen and Kaplansky showed that ifDhas exactlynnonassociate atoms, then Mn−1 is universal and that ifnis prime,M2 is universal. Thus if (D, M) is a CK domain with exactly three nonassociate atoms, M2 is universal. Also, if (D, M) is an atomic domain withM2universal, then for atomsa1, . . . , an∈D, a1· · ·anM =Mn+1 [3, Theorem 5.1]. In particular, if a and b are atoms of D, then aM =M2=bM.
Theorem3.14. Let(D, M) be a quasilocal atomic domain. Consider the three conditions: (1) D has exactly one or three nonassociate atoms, (2) D is a τ[ ]-UFD, and (3) D is SF-factorial. Then (1) ⇒ (2) ⇒ (3). If M2 is universal or D has complete integral closure Dc a DVR with [D :Dc]⊇ M, then (3)⇒(1).
Proof. (1) ⇒ (2) If D has exactly one nonassociate atom, then D is a DVR and hence a τ[ ]-UFD. So suppose thatDhas exactly three nonassociate atoms a1,a2, anda3. Then using the last line of the previous paragraph, for i 6= j, aiaj = ukak2 where k 6= i, j and uk ∈ U(D). Thus for x, y ∈ D#, [x, y] = 1 if and only if x=uai and y =vaj fori6=j and someu, v∈U(D).
Hence a∈D#is either aτ[ ]-atom or has aτ[ ]-atomic factorization of exactly
