GCD-SETS IN INTEGRAL DOMAINS 1. Introduction In this ... - CiteSeerX

Houston Journal of Mathematics c 1999 University of Houston

Volume 25, No. 1, 1999

GCD-SETS IN INTEGRAL DOMAINS D.D. ANDERSON, DAVID F. ANDERSON, JEANAM PARK

Communicated by Klaus Kaiser Abstract. Let R be an integral domain. A saturated multiplicative subset S 6= U (R) of R is a GCD-set if gcd(a, b) exists for each a, b ∈ S. We study the structure of GCD-sets of R, with emphasis on the case where R is a Dedekind domain. We show that if R is atomic, then each GCD-set is generated by completely irreducible elements, and that if R is a Dedekind domain and x is a nonzero nonunit of R, then for some N ≥ 1, xN has a completely irreducible factor. Let R be a Dedekind domain with torsion realizable pair {Cl(R), A}. If S is a GCD-set of R, then there is a subgroup GS of Cl(R) generated by an independent subset of A with Cl(R)/GS ∼ = Cl(RS ). Conversely, suppose that G is a subgroup of Cl(R) generated by an independent subset of A. Then there is a GCD-set SG of R with Cl(R)/G ∼ = Cl(RSG ).

1. Introduction In this paper, we study certain saturated multiplicative subsets of an integral domain R. We say that a saturated multiplicative set S 6= U (R) is a GCD-set if each pair of elements a, b ∈ S has a gcd(a, b) in R (and hence in S). Thus R∗ is a GCD-set if and only if R is a GCD-domain (recall that R is a GCD-domain if any two elements of R have a GCD in R [22], or equivalently, the intersection of any two principal ideals of R is principal). This paper is divided into four sections including the introduction. In section 2, we give some general results about GCD-sets. Each irreducible element in a GCD-domain is prime, while each irreducible element in a GCD-set S is completely irreducible (x is completely irreducible if the only divisors of xn (n ≥ 1) are associates of xm for 0 ≤ m ≤ n). In particular, if R is atomic (recall that R is atomic if each nonzero nonunit of R is a product of irreducible elements), then S is generated by completely irreducible elements of R. We show that each 1991 Mathematics Subject Classification.

13A15, 13F05, 13G05. 15

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GCD-set is contained in a maximal GCD-set; however, even in a GCD-domain, minimal GCD-sets need not exist. In section 3, we study completely irreducible elements, with emphasis on the case where R is a Dedekind domain. We show that if R is a Dedekind domain and x is a nonzero nonunit of R, then xN has a completely irreducible factor for some integer N ≥ 1. Thus a Dedekind domain always has a GCD-set. Also, if either Cl(R) is torsion or Cl(R) = Z, then xN is a product of completely irreducible elements for some N ≥ 1. Although the results in this paper are stated for Dedekind domains, they extend in a natural way to Krull domains. The last section investigates more deeply the structure of GCD-sets in Dedekind domains. If P is a prime ideal of a Dedekind domain R with |[P ]| < ∞, then set S[P ] = {x ∈ R∗ |xR = P1 · · · Pn with each Pi ∈ [P ]} ∪ U (R). We show that if P is nonprincipal, then S[P ] is a GCD-set if and only if [P ] contains exactly one prime ideal. Let R be a Dedekind domain with realizable pair {Cl(R), A}. If Cl(R) is torsion and S is a multiplicative subset of R, set AS = {[P ]|P n = xR for some irreducible x ∈ S and n ≥ 1} − {0} ⊆ A. Let GS be the subgroup of Cl(R) generated by AS . We show that if S is a GCD-set, then AS is an independent set, for each [P ] ∈ AS there is a unique Q ∈ [P ] such that Qn = xR for some irreducible x ∈ S and n ≥ 2, and Cl(R)/GS ∼ = Cl(RS ). Conversely, suppose that G is a subgroup of Cl(R) generated by an independent subset {gj |j ∈ J} of A. For each j ∈ J, choose a single prime ideal, say Pj , from the ideal class gj . Write n SG = h{xj |Pj j = xj R, nj = |[Pj ]|, j ∈ J}i (SG depends on the choice of the Pj ’s). We show that SG is a GCD-set and Cl(R)/G ∼ = Cl(RSG ). Hence if S 0 is a GCD-set of R, then there exists a subgroup G of Cl(R) generated by an independent subset of A such that S 0 is generated by SG and some principal primes of R. Throughout, R will denote an integral domain with quotient field K, U (R) its group of units, R∗ its set of nonzero elements, I(R) its set of irreducible elements, and X (1) (R) its set of height-one prime ideals. Also, for a nonzero ideal I of R, we denote I ∗ = I − {0}. If S is a multiplicative subset of an integral domain R, then S is generated by A ⊆ S, written hAi, if S = {ua1 · · · an |u ∈ U (R), each ai ∈ A, n ≥ 0}. The Picard group of R is denoted by Pic(R). Also, given a Krull (Dedekind) domain R, we will denote its divisor class group by Cl(R) and the class of a height-one prime ideal P in Cl(R) by [P ]. For general references on GCD-domains or their generalizations, see [8], [14], [22], [27], [1], [2], [4], or [13]. As a general reference for factorization in integral domains, see [5]. We conclude this section with several less familiar concepts which will be used throughout this paper.

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A saturated multiplicative subset S of an integral domain R is said to be a splitting multiplicative set [6], [7] (cf. [16] and [23]) if for each 0 6= r ∈ R, we can write r = sa for some s ∈ S and a ∈ R with s0 R ∩ aR = s0 aR for all s0 ∈ S. The multiplicative set T = {0 6= t ∈ R|sR ∩ tR = stR for all s ∈ S} is called the mcomplement for S. A splitting multiplicative set S of R is called an lcm-splitting multiplicative set of R if sR ∩ aR is principal for each s ∈ S, a ∈ R [7]. Let R be an atomic domain. Suppose that x1 · · · xm = y1 · · · yn ,

(∗)

where x1 , · · · , xm , y1 , · · · , yn ∈ I(R). For any nonempty subset F of I(R), let Fx = {i|xi ∈ F} and Fy = {j|yj ∈ F}. Following [10], we say that F ⊆ I(R) is a factorization set (F-set) if for any equality of the form (∗), |Fx | 6= 0 implies that |Fy | = 6 0, and that F is a half-factorial set (HF-set) if any equality of the form (∗) implies that |Fx | = |Fy |. Let G be an abelian group and C ⊆ G. We say that C is an independent set in G if n1 c1 + · · · + nk ck = 0 for ni ∈ Z and distinct ci ∈ C implies that each ni ci = 0. Most of the examples in this paper are Dedekind domains, and we will need the following definition. If for a given abelian group G and subset A ⊆ G − {0} there exists a Dedekind domain R such that Cl(R) = G and A = {g|g ∈ G and g contains a nonprincipal prime ideal of R}, then the pair {G, A} is called realizable [18], [17]. (Thus R is a UFD (PID) if and only if A = ∅.) We will use the following two results of Grams [18, Corollary 1.5 and 1.6]: (i) {Z, A} is realizable if and only if A generates Z and A contains both positive and negative integers, and (ii) if G is a torsion abelian group, then {G, A} is realizable if and only if A generates G. Let R be a Krull domain and S a multiplicative set of R. Then the natural homomorphism ϕ : Cl(R) → Cl(RS ) is surjective and kerϕ is generated by {[P ]|P ∩ S 6= ∅} (Nagata’s Theorem) [13, Corollary 7.2]. Finally, recall that if gcd(ab, ac) exists for a, b, c ∈ R∗ , then gcd(b, c) exists and gcd(ab, ac) = a · gcd(b, c) [14, Page 76, Exercise 2(a)]. 2. GCD-sets Let S be a multiplicative set of an integral domain R. We say that S satisfies Property (γ) if each pair of elements in S has a GCD in R and that S is a GCD-set of R if (i) S 6= U (R), (ii) S is saturated, and (iii) S satisfies Property (γ). Thus for a GCD-set S, gcd(s, t) ∈ S for all s, t ∈ S. A GCD-set S is called a maximal

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(resp., minimal) GCD-set if S ⊆ T (resp., T ⊆ S) and T is a GCD-set, then S = T. In this section, we give some general results about GCD-sets. Later, we will show that a Dedekind domain has a GCD-set (Theorem 3.4). However, an arbitrary integral domain need not have any GCD-sets (Remark 3.3(d)). The first two propositions include the two motivating examples of GCD-sets; namely, R∗ when R is a GCD-domain and any multiplicative set generated by principal primes. The class of GCD-domains includes UFDs and valuation domains. It is well known that a GCD-domain R is a UFD if and only if R is atomic. Proposition 2.1. Let R be an integral domain which is not a field. (a) R∗ is a GCD-set if and only if R is a GCD-domain. (b) If R is atomic, then R∗ is a GCD-set if and only if R is a UFD. Proof. (a) is clear. For (b), just note that a UFD is an atomic GCD-domain. Proposition 2.2. Let R be an integral domain and let S 6= U (R) be a saturated multiplicative set of R. (a) If S is generated by principal primes, then S is a GCD-set. (b) If S is an lcm-splitting multiplicative set of R, then S is a GCD-set and RT is a GCD-domain, where T is the m-complement for S. Proof. (a) is clear. For (b), suppose that S is an lcm-splitting multiplicative set. Then aR ∩ bR is principal for each a, b ∈ S. Thus lcm(a, b), and hence gcd(a, b), exists in R. Since S is saturated, S is a GCD-set. Also, by [7, Proposition 2.4] RT is a GCD-domain. Let R be an integral domain and let S be an lcm-splitting multiplicative set with m-complement T . We have observed that RT is a GCD-domain; however, RS is a GCD-domain if and only if R is a GCD-domain [16, Theorem 3.1]. Next, let R be a Dedekind domain with Cl(R) = Zp , p a prime. Take S = hf i, where f R = Qp for some nonprincipal Q ∈ X (1) (R). Then S is a (minimal) GCD-set (cf. Proposition 3.2(c)), and RS = Rf is a UFD (PID) by Nagata’s Theorem; so RS is a GCD-domain. However, R is not a GCD-domain (cf. Proposition 2.1). We next show that each GCD-set is contained in a maximal GCD-set. The proof of the next proposition is routine, and hence will be omitted. Proposition 2.3. Let R be an integral domain. (a) Let {Gγ } be a directed family of multiplicative subsets of R which satisfy S Property (γ). Then the multiplicative set G = Gγ also satisfies Property (γ). In particular, if each Gγ is a GCD-set, then G is also a GCD-set of R.

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(b) Let S and T be multiplicative subsets of R with S ⊆ T . If T satisfies Property (γ), then S also satisfies Property (γ). Using Zorn’s Lemma, we have the following corollary to Proposition 2.3. Corollary 2.4. Each GCD-set of an integral domain R is contained in a maximal GCD-set. Our next example shows that even in a GCD-domain, minimal GCD-sets need not exist. Example 2.5. Let R be a valuation domain with maximal ideal M . Since R is a GCD-domain, each saturated multiplicative set S 6= U (R) of R is a GCD-set. By [22, Theorem 2], S = R − P for P a nonmaximal prime ideal of R. Thus R has a minimal GCD-set if and only if M is a branched prime ideal of R, that is, there is a prime ideal Q of R directly below M . Thus if R has distinct prime S∞ ideals {Mn }∞ n=1 with {0} ⊂ M1 ⊂ · · · and M = n=1 Mn , then R does not have a minimal GCD-set (cf. [14, page 190]). However, if either R is finite dimensional or M is principal, then R has a minimal GCD-set S. In fact, if M = xR, then S = hxi. Following [21] and [26], we call a nonzero nonunit x in an integral domain R completely irreducible (or absolutely irreducible, cf. [25]) if the only divisors of xn (n ≥ 1) are associates of xm for 0 ≤ m ≤ n. Thus if x is completely irreducible, then x is irreducible. But the converse is not true (cf. Remark 3.3(c) and (d), Example 4.1(b), and Example 4.4). Clearly each prime element of R is completely irreducible. Completely irreducible elements will be studied in more detail in the next section. Let R be an integral domain and S a GCD-set of R. Thus S is also a GCDmonoid (see [15, page 52] for definitions). We call an element x ∈ R irreducible (resp., completely irreducible, prime) if x is irreducible (resp., completely irreducible, prime) as an element of R. One can also define an element x ∈ S to be irreducible (resp., completely irreducible, prime) as an element of the monoid S. Since S is saturated, an x ∈ S is irreducible (resp., completely irreducible) as an element of R if and only if it is irreducible (resp., completely irreducible) as an element of S. However, a prime element x in the monoid S need not be prime as an element of the domain R (cf. Proposition 3.2(c)). In a GCD-monoid, each irreducible element is prime [15, Theorem 6.7], and hence completely irreducible.

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Thus if R is atomic, then a GCD-set in R is just a divisor closed non-trivial factorial submonoid of R∗ . These observations give another proof of Theorem 2.6 and Proposition 3.6. We now show that each irreducible element in a GCD-set is actually completely irreducible. Theorem 2.6. Let R be an integral domain and let S be a GCD-set of R. Then each irreducible element in S is completely irreducible. In particular, if R is atomic, then S is generated by completely irreducible elements. Proof. Suppose that x ∈ S is irreducible, but not completely irreducible. Let n ≥ 2 be the smallest integer such that xn has a divisor y which is not an associate of any xm . Write xn = yz. Thus z is also not an associate of any xm , and y, z ∈ S since S is saturated. Since S is a GCD-set, gcd(xy, xn ) exists. By the minimality of n, we have that gcd(x, z) = 1. This yields gcd(xy, xn ) = y, and hence x|y, which contradicts the minimality of n. Thus each irreducible element of S is completely irreducible. The “in particular” statement is clear since S is saturated. Corollary 2.7. Let R be an integral domain and let S = hxi be a multiplicative set with x ∈ I(R). Then the following statements are equivalent. (a) S is a (minimal) GCD-set. (b) S is saturated. (c) x is completely irreducible. In particular, if R is atomic, then each minimal GCD-set of R has the form S = hxi for some completely irreducible x ∈ R. Proof. The equivalence of the three statements follows easily from the definitions; also see [12, Proposition 1.1]. The “in particular” statement now follows directly from Theorem 2.6. Let R be an integral domain. Following Sheldon [27], we say that a nonzero prime ideal P of R is a PF-prime if gcd(a, b) exists in P for each a, b ∈ P ∗ (so P ∗ satisfies Property (γ)). (Note that, unlike Sheldon, we are not assuming that R is a GCD-domain. However, by Proposition 2.8, such an R is necessarily a GCD-domain.) Recall that a nonzero ideal I of R is said to be a t-ideal if S I = It , where It = {(a1 , · · · , an )v |0 6= (a1 , · · · , an ) ⊆ I} and (a1 , · · · , an )v = ((a1 , · · · , an )−1 )−1 . In a GCD-domain, a PF-prime is the same as a prime t-ideal ([27, Theorem 2.2] and [24, Proposition 4.1]). Our next result generalizes Proposition 2.1.

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Proposition 2.8. Let R be an integral domain and let I be a nonzero ideal of R. If I ∗ satisfies Property (γ), then R is a GCD-domain. Proof. Suppose that each pair of elements in I ∗ has a GCD in R. Let a, b ∈ R∗ and x ∈ I ∗ . By assumption, gcd(ax, bx) exists; so gcd(a, b) exists in R. Thus R is a GCD-domain. The following corollary generalizes [27, Theorem 2.2]. Following Hedstrom and Houston [20], we define a (necessarily) quasilocal domain R with maximal ideal M to be a pseudo-valuation domain if whenever xy ∈ M for x, y ∈ K, then either x ∈ M or y ∈ M . Clearly a valuation domain is a pseudo-valuation domain (but not conversely). Corollary 2.9. Let R be an integral domain and let P be a nonzero prime ideal of R. Then the following statements are equivalent. (a) (b) (c) (d)

P R R R

is is is is

a a a a

PF-prime. GCD-domain and P is a t-ideal of R. GCD-domain and RP is a valuation domain. GCD-domain and RP is a pseudo-valuation domain.

Proof. (a) ⇔ (b) ⇔ (c). This follows from Proposition 2.8, [27, Theorem 2.2], and the above comments on t-ideals. (c) ⇔ (d). Suppose that R is a GCDdomain. Then RP is also a GCD-domain. Hence RP is a valuation domain if and only if RP is a pseudo-valuation domain [20, Proposition 2.2]. 3. Completely Irreducible Elements In this section, we continue our study of completely irreducible elements, with emphasis on the case where R is a Dedekind domain. Let R be a Dedekind domain and x ∈ R a nonunit. Then xR = P1e1 · · · Pnen for some distinct Pi ∈ X (1) (R) and ei > 0. Note that for zR = P1d1 · · · Pndn , z|x if and only if each di ≤ ei . Let Γx = {(a1 , · · · , an ) ∈ Zn+ | P1a1 · · · Pnan is principal}. Then Γx is a nonzero additive submonoid of Zn+ (here Z+ denotes the set of nonnegative integers). For α = (a1 , · · · , an ), β = (b1 , · · · , bn ) ∈ Γx , let α ≤ β ⇔ ai ≤ bi for each 1 ≤ i ≤ n, i.e., ≤ is the usual product order. Then Γx has only finitely many minimal elements. Note that if α, β ∈ Γx and α ≤ β, then β − α ∈ Γx , and that each nonzero element of Γx is a sum of minimal elements. Also note that for zR = P1d1 · · · Pndn , z is irreducible if and only if ζ = (d1 , · · · , dn ) ∈ Γx is minimal; and z is completely irreducible if and only if

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whenever β = (b1 , · · · , bn ) ∈ Γx is minimal and β ≤ mζ for some m ≥ 1, then ζ = β. We will also call ζ ∈ Γx completely irreducible if zR = P1d1 · · · Pndn has z completely irreducible. Our first lemma gives necessary and sufficient conditions in terms of Γx for x ∈ R to be completely irreducible. Lemma 3.1. (cf. [19, Satz 11]) Let R be a Dedekind domain and xR = P1e1 · · · Pnen , where P1 , · · · , Pn ∈ X (1) (R) are distinct and each ei > 0. Then the following statements are equivalent. (a) x is completely irreducible in R. (b) Γx = Z+ (e1 , · · · , en ). (c) (e1 , · · · , en ) is the unique minimal element of Γx . Proof. (a) ⇒ (b). Clearly Z+ (e1 , · · · , en ) ⊆ Γx . We show that Γx ⊆ Z+ (e1 , · · · , en ). Let (a1 , · · · , an ) ∈ Γx and let yR = P1a1 · · · Pnan . Since each ei > 0, there exists a least positive integer N such that ai ≤ N ei for each 1 ≤ i ≤ n. Thus xN R = yR(P1N e1 −a1 · · · PnN en −an ). Since x is completely irreducible, yR = xm R for some m with 0 ≤ m ≤ N . Thus ai = mei for each 1 ≤ i ≤ n. By the minimality of N , we have m ≥ N . This implies that yR = xN R; so (a1 , · · · , an ) = N (e1 , · · · , en ). (b) ⇒ (a). Suppose that Γx = Z+ (e1 , · · · , en ). Then (e1 , · · · , en ) is minimal in Γx , and thus x is irreducible. If y|xN , then by the unique factorization of prime ideals, yR = P1a1 · · · Pnan , where each 0 ≤ ai ≤ ei N . Thus (a1 , · · · , an ) ∈ Γx = Z+ (e1 , · · · , en ). Hence y is an associate of xm for some m with 0 ≤ m ≤ N . (b) ⇔ (c). This is clear. Proposition 3.2. Let R be an integral domain. (a) Let P be a nonzero prime ideal of R and xR = P n , where n = |[P ]| in Pic(R). Then x is completely irreducible. km (b) Suppose that x is completely irreducible and xR = P1k1 · · · Pm , where the Pi ’s are distinct prime ideals, each ki ≥ 1, and some [Pi ] has finite order. Then m = 1 and xR = P1n , where n = |[P1 ]| in Pic(R). (c) Let R be a Dedekind domain with torsion divisor class group. Then hxi is a (minimal) GCD-set if and only if x is completely irreducible, if and only if xR = P n for P a nonzero prime ideal of R with |[P ]| = n. (d) Let R be a one-dimensional quasilocal domain with maximal ideal M . If x ∈ R is completely irreducible, then x is prime. Thus x is completely irreducible if and only if R is a DVR with M = xR.

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Proof. (a) Note that P is invertible. Suppose that P l = IJ, where l ≥ 1 and I and J are integral (invertible) ideals of R. Then I 6⊆ P l+1 and J 6⊆ P l+1 ; so there are i, j ≥ 0 such that I ⊆ P i with I 6⊆ P i+1 and J ⊆ P j with J 6⊆ P j+1 . Thus I = I 0 P i and J = J 0 P j , where I 0 , J 0 6⊆ P . Then P l = I 0 J 0 P i+j yields that i + j = l and I 0 = J 0 = R. Hence I = P i and J = P j . Suppose that y|xm ; so ry = xm for some r ∈ R. Thus rRyR = xm R = P mn ; so yR = P i for some i ≥ 0. Suppose i > 0, i.e., y is a nonunit. Then n|i, say i = kn. Hence yR = P nk = xk R, and thus y is an associate of xk . Hence x is completely irreducible. km (b) Suppose that xR = P1k1 · · · Pm , where the Pi ’s are distinct primes, m > n 1, and |[P1 ]| = n. Thus P1 = yR, where y is irreducible. Then xn R = km n nkm (P1k1 · · · Pm ) = (P1n )k1 P2nk2 · · · Pm . Hence xn R ⊆ y k1 R; so y|xn . Since y is km irreducible, y is an associate of x, and hence P1k1 · · · Pm = P1n . From the first ki part of the proof of part (a) above, we get that each Pi is a power of P1 , a contradiction. Thus m = 1 and xR = P1n , where n = |[P1 ]|. (c) This follows easily from Corollary 2.7 and parts (a) and (b) above. (d) Each nonzero element of M divides some power of x [22, Theorem 108]. Thus M = xR since x is completely irreducible; so x is prime. Remark 3.3. (a) Let R be a Dedekind domain with torsion divisor class group. By Proposition 3.2(c), for each nonzero nonunit x ∈ R, xn is a product of completely irreducible elements for some n ≥ 1. (b) Let R be a Dedekind domain and x ∈ R irreducible with xR = P i Qj , where P and Q are distinct prime ideals of R with |[P ]| = |[Q]| = ∞. Then x is completely irreducible. It’s enough to show that Γx = Z+ (i, j) by Lemma 3.1. Note that (i, j) is minimal in Γx , and that Γx has no elements of the form (0, k) or (k, 0) with k 6= 0 since |[P ]| = |[Q]| = ∞. We show that (i, j) is the unique minimal element of Γx . Suppose that (a, b) ∈ Γx is minimal. Then (ai, aj), (ai, bi) ∈ Γx . Since Γx has no elements of type (0, k), k 6= 0, we must have aj = bi. Suppose a < i. Then b = ai j < j; so (i, j) is not minimal, a contradiction. Suppose a ≥ i, and hence b = ai j ≥ j. Thus (a, b) = (i, j). Hence Γx = Z+ (i, j) by Lemma 3.1. (c) Let R be a Dedekind domain with realizable pair {Z, {−3, 1, 2}}. Let A, B, and C be prime ideals of R such that [A] = −3, [B] = 1, and [C] = 2. Let xR = ABC, yR = AB 3 , and zR = A2 C 3 . Then x, y, and z are each irreducible and Γx 6= Z+ (1, 1, 1). Thus x is not completely irreducible. However, y and z are each completely irreducible by part (b) above. (d) Let F be a field, x an indeterminate, and R = F [[x2 , x3 ]]. Then R is a one-dimensional (Noetherian, and hence atomic) local domain which is not a

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DVR. By Proposition 3.2(d), R has no completely irreducible elements. Thus R has no GCD-sets by Theorem 2.6. In particular, x2 and x3 are each irreducible in R, but not completely irreducible. We next show that completely irreducible elements are rather abundant in a Dedekind domain. Remark 3.3(d) above shows that the next theorem is not true for arbitrary atomic integral domains. Theorem 3.4. Let R be a Dedekind domain and x ∈ R a nonzero nonunit. Then for some N ≥ 1, xN has a completely irreducible factor. In particular, R has a GCD-set. Proof. Let xR = P1e1 · · · Pnen for some distinct Pi ∈ X (1) (R) and ei > 0. Note that for zR = P1d1 · · · Pndn , z|x if and only if each di ≤ ei . We show by induction on n that Γx has a completely irreducible element γ = (d1 , · · · , dn ) ∈ Γx . Since each ei > 0, γ ≤ N (e1 , · · · , en ) for some N ≥ 1. Thus for zR = P1d1 · · · Pndn , z is completely irreducible and z|xN . The n = 1 case is clear, so assume n ≥ 2. Note that if Γx has a unique minimal element α, then Γx = Z+ α, and α is completely irreducible. So suppose that Γx has at least two minimal elements. We show that there is a nonzero γ = (d1 , · · · , dn ) ∈ Γx with some di = 0. Let J = {i|1 ≤ i ≤ n, and there exist α = (a1 , · · · , an ), β = (b1 , · · · , bn ) ∈ Γx with β 6= α, aj ≥ bj for 1 ≤ j ≤ i and aj = bj for some 1 ≤ j ≤ i}. Note that 1 ∈ J since Γx has at least two minimal elements. We show that if i ∈ J and 1 ≤ i < n, then i + 1 ∈ J. Let α = (a1 , · · · , an ), β = (b1 , · · · , bn ) ∈ Γx with α 6= β, aj ≥ bj for 1 ≤ j ≤ i, and aj = bj for some 1 ≤ j ≤ i. If ai+1 ≥ bi+1 , then clearly i + 1 ∈ J. Otherwise, ai+1 < bi+1 . We may assume ai+1 6= 0. Then bi+1 α, ai+1 β ∈ Γx show that i + 1 ∈ J. Hence J = {1, 2, · · · , n}; so there exist α = (a1 , · · · , an ), β = (b1 , · · · , bn ) ∈ Γx , α 6= β, with each aj ≥ bj and some ai = bi . Thus 0 6= γ = α − β ∈ Γx . Hence Γ0 = {(c1 , · · · , cn ) ∈ Γx |ci = 0} is a nonzero submonoid of Γx such that α, β ∈ Γ0 with α ≤ β ⇒ β − α ∈ Γ0 . By induction, Γ0 has a nonzero completely irreducible element π. But π is also completely irreducible in Γx . The “in particular” statement follows from Corollary 2.7. Theorem 3.5. Let R be a Dedekind domain with Cl(R) = Z and x ∈ R a nonzero nonunit. Then for some N ≥ 1, xN is a product of completely irreducible elements. Proof. Let x ∈ R be a nonzero nonunit. Since a prime element is completely irreducible, we may assume x has no prime factors. Hence xR = P1a1 · · · Prar Qb11 · · · Qbss

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with each ai , bj > 0, [Pi ] < 0, and [Qj ] > 0. The proof is by induction on n = r + s. The case n = 2 follows from Remark 3.3(b). So suppose that n ≥ 3. For some e, f > 0, yR = P1e Qf1 , where y is irreducible, in fact, completely irreducible, by Remark 3.3(b). Then either b1 e ≤ a1 f or b1 e > a1 f . First suppose that b1 e ≤ a1 f . Then xf R = P1a1 f · · · Prar f Qb11 f · · · Qbss f

and

y b1 R = P1b1 e Qb11 f .

Thus y b1 |xf , and hence xf = y b1 z for some z ∈ R such that Q1 is not in the prime ideal factorization of zR. By induction, z m is a product of completely irreducible elements for some m ≥ 1. Thus xf m is a product of completely irreducible elements. Next, if b1 e > a1 f , then the prime ideal factorizations of xe R and y a1 R give y a1 |xe ; so xe = y a1 w for some w ∈ R. Now, P1 is not in the prime ideal factorization of wR. So wl is a product of completely irreducible elements for some l ≥ 1 by induction. Hence xN is a product of completely irreducible elements for some N ≥ 1. Remark 3.3(a) and the preceding theorem motivate the following question. Question. Let R be a Dedekind domain and x ∈ R a nonzero nonunit. Is xN a product of completely irreducible elements for some integer N ≥ 1 ? The final results of this section show that in the Dedekind domain with torsion divisor class group setting, completely irreducible elements behave very much like prime elements. Proposition 3.6 generalizes [21, Proposition 1(iii)]. Proposition 3.6. ([19, Satz 12]) Let R be a Dedekind domain with torsion divisor class group. Then any factorization of a nonzero nonunit of R into a product of completely irreducible elements is unique (up to order and associates). In particular, this unique factorization holds for all nonunits in a GCD-set of R. Proof. This follows from the unique factorization of prime ideals in R since each completely irreducible element of R has the form xR = P n for some prime ideal P of R by Proposition 3.2(b). The “in particular” statement follows from the above remarks and Theorem 2.6. Corollary 3.7. Let R be a Dedekind domain with torsion divisor class group. Then every irreducible element of R is completely irreducible if and only if R is a UFD (PID). Our next example shows that the torsion divisor class group hypothesis is needed in both Proposition 3.6 and Corollary 3.7.

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Example 3.8. Let R be a Dedekind domain with realizable pair {Z, {−1, 1}}. Then each nonzero ideal class contains infinitely many prime ideals [17, Theorem 8]. Let A, B, C, and D be distinct prime ideals of R with [A] = [B] = −1 and [C] = [D] = 1. Let a, b, c, and d ∈ I(R) be defined by aR = AC, bR = BD, cR = BC, and dR = AD. Then a, b, c, and d are all completely irreducible by Remark 3.3(b). However, we have ab = ucd for some unit u ∈ R; so unique factorization as products of completely irreducible elements fails. It is interesting to note that each irreducible element of R is in fact completely irreducible (since each irreducible x has the form xR = P Q with [P ] = −1 and [Q] = 1). 4. GCD-sets In Dedekind Domains In this section, we investigate more deeply the structure of GCD-sets in Dedekind domains. Let R be a Dedekind domain and P ∈ X (1) (R). Following [10], we set [ HP = P ∩ I(R) and H[P ] = {HQ |Q ∈ X (1) (R) and Q ∈ [P ]}. Then HP and H[P ] are each F-sets [10, Theorem 1.7]. If R is a Dedekind domain and P is a nonzero prime ideal of R with finite order in Cl(R), then as in [11], we set S[P ] = {x ∈ R∗ | xR = P1 · · · Pn with each Pi ∈ [P ]} ∪ U (R). It is easily verified that S[P ] is a saturated multiplicative set of R with S[P ] ∩ I(R) ⊆ H[P ] . In particular, if P is principal, then S[P ] ∩ I(R) is the set of all prime elements of R (cf. Corollary 4.3). Let R be a Dedekind domain with torsion divisor class group. It is known that S[P ] ∩ I(R) = H[P ] if and only if H[P ] is an HF-set [10, Corollary 3.10], if and only if S[P ] is a splitting multiplicative set [11, Theorem 3.8]. Thus, if S = S[P ] is a splitting multiplicative set, then S is generated by the HF-set H[P ] . Moreover, S[P ] is a splitting multiplicative set for each P ∈ X (1) (R) if and only if A is independent, where {Cl(R), A} is a realizable pair [11, Theorem 3.8]. In this case, each overring of R is a half-factorial domain [9, Theorem 2.5], [11, Theorem 3.7]. (Recall that an atomic domain R is a half-factorial domain (HFD) if any two factorizations of a given nonzero nonunit as products of irreducibles have the same length; cf. [28], [5].) Example 4.1. (a) Let D be a Dedekind domain with Cl(D) = Z2 and let T be the multiplicative set generated by all principal primes of D. Then R = DT has no principal primes and Cl(R) = Z2 . Thus R∗ = S[P ] for any P ∈ X (1) (R), and hence S[P ] is not a GCD-set for any P ∈ X (1) (R) by Proposition 2.1 (cf. Corollary 4.5).

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(b) Let R be a Dedekind domain with Cl(R) = Z2 . For a nonprincipal prime ideal P , let P 2 = yR. Then y is irreducible and P = (x, y) for some irreducible x ∈ P − P 2 . Thus we have that x2 ∈ P 2 = yR, and hence x2 = yz for some irreducible z since R is an HFD ([28, Theorem 1.4]). It is easy to show that gcd(xy, yz) does not exist and that x is not completely irreducible. We next determine when S[P ] is a GCD-set. Theorem 4.2. Let R be a Dedekind domain and let P be a nonprincipal prime ideal of R with |[P ]| = n < ∞. Then the following statements are equivalent. (a) (b) (c) (d) (e) (f)

S[P ] is a GCD-set. Each x ∈ S[P ] ∩ I(R) is P -primary. [P ] contains exactly one prime ideal. S S[P ] = R − {Q ∈ X (1) (R)|Q 6= P }. S[P ] = hxi, where xR = P n . S[P ] is a minimal GCD-set.

Proof. Clearly we have (b) ⇒ (c), (c) ⇒ (d), (d) ⇒ (e), and (f ) ⇒ (a). (a) ⇒ (b). Suppose that S[P ] is a GCD-set. Let x ∈ S[P ] ∩ I(R). By Theorem 2.6, x is completely irreducible; so xR = P1n is primary for some P1 ∈ [P ] by Proposition 3.2(b). Let zR = P1 P n−1 . Then z ∈ S[P ] is also completely irreducible. Thus P1 = P by Proposition 3.2(b). Hence x is P -primary. (e) ⇒ (f ). Suppose that S[P ] = hxi, where xR = P n with n = |[P ]|. Thus x is completely irreducible by Proposition 3.2(a); so S[P ] is a minimal GCD-set by Corollary 2.7. Corollary 4.3. Let R be a Dedekind domain with torsion divisor class group and let P be a nonzero prime ideal of R. Then the following statements are equivalent. (a) (b) (c) (d) (e)

hH[P ] i is a GCD-set. hH[P ] i satisfies Property (γ). S[P ] is a GCD-set and H[P ] is an HF-set. P is principal. hH[P ] i = S[P ] is the multiplicative set generated by all principal primes of R

Proof. (a) ⇒ (b) is clear by definition. For (b) ⇒ (c), suppose that hH[P ] i satisfies Property (γ). Let x ∈ H[P ] and suppose that xR = P1 · · · Pk Q1 · · · Qt , where each Pi ∈ [P ], each Qj ∈ / [P ], and k ≥ 1, t ≥ 0. Let I = Q1 · · · Qt , |[P ]| = n, |[I]| = m, and let l = lcm(n, m) with l = nb = ma. Now assume that P1 6= P2 .

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Then xl R = P1l · · · Pkl I l = xb1 · · · xbk y a R, where each xi R = Pin and yR = I m . Now, x, xb1 ∈ hH[P ] i. By assumption, gcd(x, xb1 ) exists and equals 1 since P1 6= P2 . Since x and xb1 each divide xl and gcd(xl , xl−1 xb1 ) = xl−1 , we have xb1 |xl−1 , and hence xxb1 |xl . This yields xxb1 f R = xl R for some f ∈ R. But this contradicts the unique factorization of prime ideals in the Dedekind domain R. Thus P1 = P2 = · · · = Pk ; so xR = P1k Q1 · · · Qt , t ≥ 0. Claim. t = 0. a Proof of Claim. Notice that xl = uxbk 1 y for some unit u of R. Since bk l a gcd(xx1 , x ) exists, we have that gcd(x, y ) exists. Since gcd(x, y a ) = 1, l bk bk gcd(xxbk 1 , x ) = x1 . Now, if t ≥ 1, then x does not divide x1 , a contradiction. Hence t = 0. Thus H[P ] ⊆ S[P ] ∩ I(R); so H[P ] = S[P ] ∩ I(R). This yields H[P ] is an HF-set by [10, Corollary 3.10], [11, Theorem 3.8]. Note that hH[P ] i = S[P ]. It follows that S[P ] satisfies Property (γ). Since S[P ] is saturated, S[P ] is a GCD-set. For (c) ⇒ (d), suppose that S[P ] is a GCD-set and H[P ] is an HF-set, but P is not principal. By Theorem 4.2, H[P ] = HP ; so HP is also an HF-set. Hence P is principal [10, Proposition 3.4], a contradiction. Thus P must be principal. For (d) ⇒ (e), note that if P is principal, then hH[P ] i = S[P ] is the multiplicative set generated by all principal primes of R. (e) ⇒ (a) follows from Proposition 2.2(a).

Example 4.4. ([11, Example 5(2)]) Let R be a Dedekind domain with realizable pair {Z2 ⊕ Z4 , A}, where A = {(1, 1), (1, 2), (0, 1)}. Let P1 be a prime ideal of class (1, 1), P2 a prime ideal of class (1, 2), and P3 a prime ideal of class (0, 1). Let xR = P14 , yR = P22 , zR = P34 , and αR = P1 P2 P3 . Then x, y, z, and α are irreducible in R and α4 = uxy 2 z for some unit u of R. Let S = hH[P1 ] i. Then S is not a saturated multiplicative subset of R since α ∈ S, but y ∈ / S. However, hI(R) − H[P1 ] i is a saturated multiplicative set [10, Theorem 1.7], [12, Theorem 1.5]. Note that αx, α4 ∈ S and gcd(αx, α4 ) does not exist. To show this, suppose that gcd (αx, α4 ) exists in R. Since gcd(x, α3 ) = 1, gcd(αx, α4 ) = α. Now, x|αx and x|α4 imply that x|α. But this contradicts the unique factorization of prime

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ideals in the Dedekind domain R. Thus gcd(αx, α4 ) does not exist. Hence S does not satisfy Property (γ). This also shows that α is not completely irreducible. Corollary 4.5. Let R be a Dedekind domain with torsion divisor class group {Cl(R), A}. If there exists a nonprincipal prime ideal P of R such that S[P ] is a GCD-set, then A is not an independent set of Cl(R). Proof. Suppose that P is a nonprincipal prime ideal of R such that S[P ] is a GCD-set and that A is independent. By [11, Theorem 3.8], H[P ] is an HFset. Thus P is principal by Corollary 4.3, a contradiction, and hence A is not independent. Example 4.6. Let R be a Dedekind domain with realizable pair {G, A}, where L G= Zni and A = {(1, 0, 0, · · · ), (0, 1, 0, · · · ), · · · }. Then A is an independent subset of Cl(R). Thus Cl(R) has no nonzero element containing exactly one prime ideal by Theorem 4.2 and Corollary 4.5 (cf. [17, Theorem 8]). Let R be a Dedekind domain with divisor class group G. Following [17], let ∆(g) ∈ {0, 1, 2, · · · } ∪ {∞} denote the number of prime ideals of R in the class g ∈ G. Let G be a countably generated abelian group generated by A as a monoid. Suppose that A = B ∪ C is a partition of A such that B 0 ∪ C generates G as a monoid for each cofinite subset B 0 of B. Then there exists a Dedekind domain R such that {G, A} is a realizabe pair, ∆(c) = ∞ for all c ∈ C, and ∆(b) is an arbitrarily specified positive integer for each b ∈ B [17, Theorem 8]. In particular, if G is torsion, then there exists a Dedekind domain R such that {G, A} is realizable, ∆(b) = 1 for each b ∈ B and ∆(c) = ∞ for each c ∈ C [17, Theorem 8]. By Theorem 4.2, S[P ] is a (minimal) GCD-set for each [P ] ∈ B. Moreover, if B is independent, then h{S[P ]|[P ] ∈ B}i is a GCD-set (see Corollary 4.12). Let R be a Dedekind domain with torsion divisor class group Cl(R). If Rf is a UFD (PID) for each nonzero nonunit f ∈ R, then Cl(R) is cyclic [3, Theorem 4.6]. We next give another case when Cl(R) is cyclic. Proposition 4.7. Let R be a Dedekind domain and let P be a prime ideal of R with |[P ]| < ∞. If RS[P ] is a UFD (PID), then Cl(R) = h[P ]i is cyclic. Proof. Let S = S[P ]. Then the kernel of the natural epimorphism ϕ : Cl(R) → Cl(RS ) is h[P ]i by Nagata’s Theorem. Thus Cl(R) = h[P ]i is cyclic.

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Let R be a Dedekind domain with torsion divisor class group {Cl(R), A}, S a multiplicative subset of R, and AS = {[P ]|P n = xR for some irreducible x ∈ S and n ≥ 1} − {0} ⊆ A. Let GS be the subgroup of Cl(R) generated by AS . We next show that if S is a GCD-set, then AS is an independent subset of Cl(R). Theorem 4.8. With the notation above, if S is a GCD-set of R, then AS is an independent set, for each [P ] ∈ AS there is a unique Q ∈ [P ] with Qn = xR for some irreducible x ∈ S and n ≥ 2, and Cl(R)/GS ∼ = Cl(RS ). Proof. Suppose that S is a GCD-set. By Theorem 2.6, each irreducible x in S is completely irreducible. Thus there exists a unique prime ideal Px of R such that xR = Pxnx , where nx = |[Px ]|. By definition, GS is generated by AS = {[Px ]| x ∈ S and x is irreducible in S} − {0}. Suppose now that AS is not independent. Then there exists an irreducible y ∈ R such that yR = Px1 · · · Pxk , where [Px1 ] 6= [Px2 ] and each xi ∈ S is irreducible. Since each [Pxi ] has finite order and S is saturated, we have y ∈ S, a contradiction since y is not completely irreducible. Hence GS is generated by the independent set AS . The proof of (a) ⇒ (b) of Theorem 4.2 shows that [Px ] 6= [Py ] if x and y aren’t associates. By Nagata’s Theorem, κ = ker(Cl(R) → Cl(RS )) is generated by the classes of the prime ideals which meet S. Let P be a prime ideal which meets S. By Theorem 2.6, P contains a completely irreducible element, say x ∈ S. Clearly xR = P n , where n = |[P ]|. Thus [P ] ∈ GS , and hence κ ⊆ GS . Conversely, let P be a prime ideal with P n = yR for some irreducible y ∈ S and n ≥ 1. Then y ∈ P , so P ∩ S 6= ∅. Hence GS ⊆ κ, and we have the equality. Thus Cl(R)/GS ∼ = Cl(RS ) by Nagata’s Theorem. Remark 4.9. Let R be a Dedekind domain with torsion divisor class group Cl(R). For a subset P ⊆ X (1) (R), let V (P) be the set of all x ∈ R∗ having all its prime divisors in P. Then S is a saturated multiplicative set if and only if S is of the form V (P) ∪ U (R) for some P ⊆ X (1) (R) [19, Proposition 3]. If S is a GCD-set, then {[P ] | P ∈ P, P nonprincipal} is an independent set of Cl(R). By Theorem 4.8, it’s enough to show that {[P ] | P ∈ P, P nonprincipal} = AS . Now, we may assume that S has no principal primes. If [P ] ∈ AS , then P ∈ P by the unique factorization of prime ideals in Dedekind domains. Suppose that P ∈ P. Then P n = xR, where n = |[P ]|. Since x ∈ S = V (P) ∪ U (R) is irreducible, [P ] ∈ AS . Hence {[P ] | P ∈ P, P nonprincipal} = AS . Conversely,

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if P ⊆ X (1) (R) and each nonprincipal P ∈ P is the unique prime in [P ], then S = V (P) ∪ U (R) is a GCD-set (see Corollaries 4.11 and 4.12). However, {[P ] | P ∈ P, P nonprincipal} independent does not imply that S is a GCD-set. For a nonprincipal prime P , S[P ] = V (P) ∪ U (R), where P = {Q ∈ X (1) (R) | Q ∈ [P ]}. By Theorem 4.2, for P nonprincipal, S[P ] is a GCD-set if and only if P = {P }, a singleton set. Now, let P ⊆ X (1) (R). The proof of (4) ⇒ (1) [11, Theorem 3.8] shows that if S[P ] is a splitting multiplicative set for each P ∈ P, then {[P ] | P ∈ P, P nonprincipal} is an independent set of Cl(R). Let R be a Dedekind domain with torsion divisor class group {Cl(R), A}. Now, suppose that S is a GCD-set. By Theorem 4.8, there exists a subgroup GS generated by the independent subset AS of A such that Cl(R)/GS ∼ = Cl(RS ). Conversely, suppose that G is a subgroup of Cl(R) generated by an independent subset {gj |j ∈ J} of A. For each j ∈ J, choose a single prime ideal, say Pj , from the ideal class gj . Write nj

SG = h{xj |Pj

= xj R, nj = |[Pj ]|, j ∈ J}i.

(Note that SG depends on the choice of the Pj ’s and SG = U (R) if J = ∅.) The next theorem shows that SG is a GCD-set and Cl(R)/G ∼ = Cl(RSG ). Theorem 4.10. Let R be a Dedekind domain with torsion divisor class group {Cl(R), A}. If G is a subgroup of Cl(R) generated by an independent subset of A, then there is a GCD-set SG such that Cl(R)/G ∼ = Cl(RSG ). Proof. Let {gj |j ∈ J} be an independent subset of A which generates G. For each j ∈ J, choose exactly one prime ideal, say Pj , from the ideal class gj . Let S n SG = h{xj |Pj j = xj R, nj = |[Pj ]|, j ∈ J}i. Since HQ is an F-set, where the union is taken over all Q ∈ / {Pj |j ∈ J}, we have that D E [ I(R) − HQ S is saturated [10, Page 95], [12, Theorem 1.5]. For each j ∈ J, xj ∈ / HQ , where S nj xj R = Pj with nj = |[Pj ]|. This yields SG ⊆ hI(R) − HQ i. Suppose now S that y ∈ I(R) − HQ . We may assume that yR = P1 · · · Pn . Since {[Pj ]|j ∈ J} is independent, [P1 ] = · · · = [Pn ], so yR = Pjn for some j ∈ J, and hence y ∈ SG . Thus D E [ SG = I(R) − HQ .

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Hence SG is saturated. Thus each divisor of xn1 1 · · · xnk k ∈ SG is an associate of mk 1 some xm with 0 ≤ mi ≤ ni by the unique factorization of prime ideals 1 · · · xk (cf. Proposition 3.6). This also shows that SG satisfies Property (γ), and hence SG is a GCD-set. Clearly G = GSG ; so Cl(R)/G ∼ = Cl(RSG ) by Theorem 4.8. Let SG be as above and let P be any set of principal primes of R. Then S = hSG ∪ Pi is clearly also a GCD-set of R. Conversely, if S 0 is a GCD-set of R, then there exists a subgroup G of Cl(R) generated by an independent subset of A such that S 0 is generated by SG and some set P of principal primes of R. The next corollary completely characterizes GCD-sets in a Dedekind domain R with torsion divisor class group. 0

Corollary 4.11. Let R be a Dedekind domain with torsion divisor class group and let S 6= U (R) be a multiplicative subset of R. Then S is a GCD-set if and only if S = hG ∪ Pi, where P is a set of principal primes of R and G ⊆ I(R) such that for each x ∈ G there is a Px ∈ X (1) (R) and nx ≥ 2 with Pxnx = xR, [Px ] 6= [Py ] for x 6= y, and {[Px ]|x ∈ G} is independent. Corollary 4.12 follows from Theorem 4.10 and Corollary 4.5. Corollary 4.12. Let R be a Dedekind domain with torsion divisor class group {Cl(R), A}. If B ⊆ A is an independent subset of Cl(R) and each b ∈ B contains exactly one prime ideal, then h{S[P ] | [P ] ∈ B}i is a GCD-set. In particular, if B = A, then R is a UFD (PID). Acknowledgements. This research was conducted while the third author visited The University of Tennessee at Knoxville. He gratefully acknowledges the hospitality of The University of Tennessee at Knoxville. His work was supported in part by funds from the Basic Science Research Institute Program, Korea Research Foundation, 1998-015-D00017. References [1] D.D. Anderson, π-domains, overrings, and divisorial ideals, Glasgow Math. J. 19 (1978), 199–203. [2] D.D. Anderson and D.F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Paul. 28 (1979), 215–221. [3] D.D. Anderson and D.F. Anderson, Locally factorial integral domains, J. Algebra 90 (1984), 265–283. [4] D.D. Anderson and D.F. Anderson, Some remarks on star operations and the class group, J. Pure Appl. Algebra 51 (1988), 27–33. [5] D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19.

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[6] D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78–93. [7] D.D. Anderson, D.F. Anderson and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37. [8] D.D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Ital. A(7) 8 (1994), 397–402. [9] D.F. Anderson, S.T. Chapman and W.W. Smith, Overrings of half-factorial domains, II, Comm. Algebra 23 (1995), 3961–3976. [10] D.F. Anderson, S.T. Chapman and W.W. Smith, Factorization sets and half-factorial sets in integral domains, J. Algebra 178 (1995), 92–121. [11] D.F. Anderson and J. Park, Locally half-factorial domains, Houston J. Math. 23 (1997), 617–630. [12] D.F. Anderson, J. Park, G. Kim and H. Oh, Splitting multiplicative sets and elasticity, Comm. Algebra 26 (1998), 1257–1276. [13] R.M. Fossum, The divisor class group of a Krull domain, Springer-Verlag, New York, 1973. [14] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972. [15] R. Gilmer, Commutative semigroup rings, Univ. of Chicago Press, Chicago, 1984. [16] R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974), 65–86. [17] R. Gilmer, W. Heinzer and W.W. Smith, On the distribution of prime ideals within the ideal class group, Houston J. Math. 22 (1996), 51–59. [18] A. Grams, The distribution of prime ideals of a Dedekind domain, Bull. Austral. Math. Soc. 11 (1974), 429–441. [19] F. Halter-Koch, Halbgruppen mit Divisorentheorie, Exposition. Math. 8 (1990), 27–66. [20] J. Hedstrom and E. Houston, Pseudo-valuation domains, Pacific J. Math. 75 (1978), 137– 147. [21] J. Kaczorowski, A pure arithmetical characterization for certain fields with a given class group, Colloq. Math. 45 (1981), 327–330. [22] I. Kaplansky, Commutative rings, rev. ed., Univ. of Chicago Press, Chicago, 1974. [23] J.L. Mott and M. Schexnayder, Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284 (1976), 388–401. [24] J.L. Mott and M. Zafrullah, On Pr¨ ufer v-multiplication domains, Manuscripta Math. 35 (1981), 1–26. [25] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer-Verlag, New York, 1990. [26] D. Rush, An arithmetic characterization of algebraic number fields with a given class group, Math. Proc. Camb. Phil. Soc. 94 (1983), 23–28. [27] P. Sheldon, Prime ideals in GCD-domains, Canad. J. Math. 26 (1974), 98–107. [28] A. Zaks, Half-factorial domains, Israel J. Math. 137 (1980), 281–302.

Received June 11, 1998

(D.D. Anderson) Dept. of Mathematics, The University of Iowa, Iowa City, IA 52242 E-mail address: [email protected]

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(D.F. Anderson) Dept. of Mathematics, The University of Tennessee, Knoxville, TN 37996 E-mail address: [email protected] (J. Park) Dept. of Mathematics, Inha University, Inchon, Korea, 402-751 E-mail address: [email protected]