On S-strong Mori domains

Journal of Algebra 416 (2014) 314–332

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Journal of Algebra www.elsevier.com/locate/jalgebra

On S-strong Mori domains Hwankoo Kim a , Myeong Og Kim b , Jung Wook Lim b,∗ a

Department of Information Security, Hoseo University, Asan 336-795, Republic of Korea b Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

a r t i c l e

i n f o

Article history: Received 12 March 2014 Available online 15 July 2014 Communicated by Kazuhiko Kurano MSC: 13A15 13B25 13E99 13F05 13G05 Keywords: S-w-finite S-w-principal S-strong Mori domain S-factorial domain S-strong Mori module Krull domain

a b s t r a c t Let D be an integral domain, S be a (not necessarily saturated) multiplicative subset of D, w be the so-called w-operation on D, and M be a unitary D-module. As generalizations of strong Mori domains (respectively, UFDs) and strong Mori modules, we define D to be an S-strong Mori domain (respectively, S-factorial domain) if for each nonzero ideal I of D, there exist an s ∈ S and a w-finite type (respectively, principal) ideal J of D such that sI ⊆ J ⊆ Iw ; and M to be an S-strong Mori module if M is a w-module and for each nonzero submodule N of M , there exist an s ∈ S and a w-finite type submodule F of N such that sN ⊆ F ⊆ Nw . This paper presents some properties of S-strong Mori domains, S-factorial domains and S-strong Mori modules. © 2014 Elsevier Inc. All rights reserved.

0. Introduction Throughout this paper, D is an integral domain and S is a (not necessarily saturated) multiplicative subset of D. In [5], the authors introduced the concept of “almost * Corresponding author. E-mail addresses: [email protected] (H. Kim), [email protected] (M.O. Kim), [email protected] (J.W. Lim). http://dx.doi.org/10.1016/j.jalgebra.2014.06.015 0021-8693/© 2014 Elsevier Inc. All rights reserved.

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finitely generated” to study Querre’s characterization of divisorial ideals in integrally closed polynomial rings. Later, Anderson and Dumitrescu [3] abstracted this notion to any commutative ring and introduced the concepts of S-Noetherian rings and S-principal ideal rings. Let R denote a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. Then R is called an S-Noetherian ring (respectively, S-principal ideal ring (S-PIR)) if each ideal of R is S-finite (respectively, S-principal), i.e., for each ideal I of R, there exist an s ∈ S and a finitely generated (respectively, principal) ideal J of R such that sI ⊆ J ⊆ I. In [3], Anderson and Dumitrescu tied together several different results which also generalize well-known results on Noetherian rings. Later, Liu studied in [17] when the generalized power series ring is S-Noetherian. Recently, Lim and Oh investigated S-Noetherian properties on composite ring extensions and amalgamated algebras along an ideal in [15] and [16], respectively. The main purpose of this article is to introduce and to investigate the notions of S-strong Mori domains, S-strong Mori modules and S-factorial domains. In fact, an S-strong Mori domain (respectively, S-factorial domain) generalizes both an S-Noetherian domain (respectively, S-principal ideal domain) and a strong Mori domain (respectively, UFD); and an S-strong Mori module generalizes a strong Mori module. (Definitions of S-strong Mori domains and S-factorial domains (respectively, S-strong Mori modules) will be given in Section 1 (respectively, Section 2).) This paper consists of four sections including introduction. In Section 1, we study basic properties of S-strong Mori domains and S-factorial domains. We show that for any maximal w-ideal M of D, D is a (D \ M )-strong Mori domain (respectively, (D \ M )-factorial domain) if and only if DM is a Noetherian domain (respectively, PID) and for every nonzero finitely generated (respectively, principal) ideal J of D, Jw DM ∩ D = Jw : s for some s ∈ D \ M (Proposition 1.4). In Section 2, we study more properties of S-strong Mori domains, which generalize well-known facts on strong Mori domains. More precisely, we prove that if S is an anti-archimedean subset of D, then D is an S-strong Mori domain if and only if the polynomial ring D[X] is an S-strong Mori domain, if and only if the t-Nagata ring D[X]Nv is an S-strong Mori domain, if and only if D[X]Nv is an S-Noetherian domain (Theorems 2.8 and 2.10). We also show that if L is an S-strong Mori submodule of a w-module M such that M/L is an S-strong Mori module, then M is an S-strong Mori module; and if D is an S-strong Mori domain and M is an S-w-finite w-module as a D-module, then M is an S-strong Mori module (Theorem 2.11). Finally, Section 3 is devoted to study the Cohen type theorem for S-factorial domains and to give new characterizations of Krull domains. In fact, we show that D is an S-factorial domain if and only if every prime w-ideal of D (disjoint from S) is S(-w)-principal (Theorem 3.2); and D is a Krull domain if and only if D is a (D \ M )-factorial domain for all maximal w-ideals M of D (Proposition 3.6). In order to help the reader’s better understanding, we review some definitions and notation. Let D be an integral domain with quotient field K, and let F(D) be the set of nonzero fractional ideals of D. For an I ∈ F(D), set I −1 := {x ∈ K | xI ⊆ D}. The mapping on F(D) defined by I → Iv := (I −1 )−1 is called the v-operation on D;

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 the mapping on F(D) defined by I → It := {Jv | J is a nonzero finitely generated fractional subideal of I} is called the t-operation on D; and the mapping on F(D) defined by I → Iw := {x ∈ K | xJ ⊆ I for some finitely generated ideal J of D such that Jv = D} is called the w-operation on D. It is easy to see that I ⊆ Iw ⊆ It ⊆ Iv for all I ∈ F(D). An I ∈ F(D) is a v-ideal (or divisorial ideal) (respectively, t-ideal, w-ideal) if Iv = I (respectively, It = I, Iw = I). Clearly, if an I ∈ F(D) is finitely generated, then Iv = It ; and a height-one prime ideal is a t-ideal. For ∗ = v, t or w, a maximal ∗-ideal means a ∗-ideal which is maximal among proper integral ∗-ideals. It was shown in [2, Corollary 2.17] that the notion of maximal t-ideals coincides with that of maximal w-ideals. Let w-Max(D) be the set of maximal w-ideals of D. It is well known that w-Max(D) = ∅ if D is not a field; and a maximal w-ideal is a prime ideal. A w-ideal I ∈ F(D) is of w-finite type if I = Jw for some finitely generated ideal J of D. An I ∈ F(D) is said to be invertible (respectively, w-invertible) if II −1 = D (respectively, (II −1 )w = D). We say that D has finite t-character (respectively, finite w-character) if every nonzero nonunit element in D is contained in only finitely many maximal t-ideals (respectively, maximal w-ideals) of D. Since the notion of maximal t-ideals coincides with that of maximal w-ideals, the concept of finite t-character is precisely the same as that of finite w-character. Recall that an ideal J of D is a Glaz–Vasconcelos ideal (GV-ideal) if J is finitely generated and J −1 = D. Let GV(D) be the set of GV-ideals of D. Then Iw = {x ∈ K | Jx ⊆ I for some J ∈ GV(D)} for all I ∈ F(D). Let M be a (not necessarily torsion-free) module over an integral domain D and let ED (M ) denote the injective envelope (or injective hull) of M . If no confusion arises, we write E(M ) for ED (M ). Set r(M ) := {x ∈ M | (annD (x))w = D}. Following [13], the w-closure of M is defined by Mw = p−1 (r(E(M )/M )), where p : E(M ) → E(M )/M is the canonical projection. Then it is easy to see that Mw = {x ∈ E(M ) | Jx ⊆ M for some J ∈ GV(D)}, Mw is independent of E(M ) (up to isomorphism), and M ⊆ Mw ⊆ E(M ). We say that M is a w-module (or semidivisorial) if M = Mw . Also, M is w-finite type if M is a w-module and M = Bw for some finitely generated submodule B of M . Any undefined terminology used in this article is standard as in [6,12] or will be explained in the course of this paper. 1. Basic results Let D be an integral domain and S a (not necessarily saturated) multiplicative subset of D. We say that a nonzero ideal I of D is S-w-finite (respectively, S-w-principal) if there exist an s ∈ S and a w-finite type (respectively, principal) ideal J of D such that sI ⊆ J ⊆ Iw . We also define D to be an S-strong Mori domain (S-SMdomain) (respectively, S-factorial domain (or S-unique factorization domain)) if each nonzero ideal of D is S-w-finite (respectively, S-w-principal). Clearly, an S-finite (respectively, S-principal) ideal is S-w-finite (respectively, S-w-principal); so S-Noetherian domains (respectively, S-principal ideal domains) are always S-SM-domains (respectively, S-factorial domains). Note that a nonzero ideal I is S-w-finite (respectively,

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S-w-principal) if and only if Iw is S-w-finite (respectively, S-w-principal); so D is an S-SM-domain (respectively, S-factorial domain) if and only if every w-ideal of D is S-w-finite (respectively, S-w-principal). Remark 1.1. (1) Let I be an S-w-finite ideal of D. Then there exist an s ∈ S and a finitely generated ideal J of D such that sI ⊆ Jw ⊆ Iw . Since J is finitely generated, we can find a Q ∈ GV(D) such that QJ ⊆ I (cf. [23, Lemma 1.1]). Note that (QJ)w = Jw and QJ is finitely generated. Hence we may assume that J is a finitely generated subideal of I by replacing J with QJ. Thus we can conclude that D is an S-SM-domain if and only if for each nonzero ideal I of D, there exist an s ∈ S and a finite generated subideal J of I such that sI ⊆ Jw ⊆ Iw . (2) Let I be a nonzero ideal of D. If I is S-w-principal, then there exist an s ∈ S and a principal ideal (d) such that sI ⊆ (d) ⊆ Iw ; so sIw ⊆ (d) ⊆ Iw . Hence Iw is S-principal. Conversely, if Iw is S-principal, then we can find a t ∈ S and a principal ideal (e) of D such that tIw ⊆ (e) ⊆ (Iw )w = Iw ; so tI ⊆ (e) ⊆ Iw . Hence I is S-w-principal. Thus D is an S-factorial domain if and only if every w-ideal of D is S-principal. (3) Let I be an S-w-principal ideal of D. Then there exist an s ∈ S and a principal ideal (d) such that sI ⊆ (d) ⊆ Iw . If d can be chosen in I, then sI ⊆ (d) ⊆ I; so I is S-principal. Before investigating S-w-properties, we first study how w-ideals behave in an integral domain D and its quotient ring DS . While the proofs can be obtained by [20, Theorem 6.7.10], [27, Theorem 8], [22, Proposition 1.2] and [21, Lemma 2.6(1)], we include them for the sake of completeness. Lemma 1.2. Let D be an integral domain, S a multiplicative subset of D and I a nonzero ideal of D. Then the following assertions hold. (1) If I is a w-ideal of D, then IDS ∩ D is a w-ideal of D. (2) If IDS is a w-ideal of DS , then IDS ∩ D is a w-ideal. (3) Iw DS ⊆ (IDS )w and (Iw DS )w = (IDS )w . Proof. (1) Let a ∈ (IDS ∩D)w . Then there exists a J ∈ GV(D) such that Ja ⊆ IDS ∩D. Since J is finitely generated, Jsa ⊆ I for some s ∈ S; so sa ∈ Iw = I. Hence a ∈ IDS ∩D, and thus IDS ∩ D is a w-ideal of D. (2) Let a ∈ (IDS ∩ D)w . Then we can find a GV-ideal J of D such that Ja ⊆ IDS ∩ D. Note that JDS ∈ GV(DS ) (cf. [10, Lemma 3.4(1)]); so aJDS ⊆ IDS . Hence a ∈ (IDS )w = IDS , and thus a ∈ IDS ∩ D. (3) Let a ∈ Iw DS . Then sa ∈ Iw for some s ∈ S; so Jsa ⊆ I for some J ∈ GV(D). Hence aJDS ⊆ IDS , and thus a ∈ (IDS )w . The second assertion follows because by (2), (Iw DS )w ⊆ ((IDS )w )w = (IDS )w ⊆ (Iw DS )w . 2

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Let f : D → E be a ring homomorphism. We say that every nonzero ideal of E is extended from D with respect to f if every nonzero ideal I of E is of the form I = f (A)E for some nonzero ideal A of D. Recall that an integral domain D is a strong Mori domain (SM-domain) if it satisfies the ascending chain condition on integral w-ideals of D, or equivalently, every w-ideal of D is of w-finite type [23, Theorem 4.3]. Clearly, if D is an SM-domain (respectively, UFD), then D is an S-SM-domain (respectively, S-factorial domain). The next proposition collects some properties of our S-w-concepts. Proposition 1.3. Let S be a multiplicative subset of an integral domain D. Then the following statements hold. (1) Let I be a w-ideal of D. If IDS ∩ D is S-w-finite (respectively, S-w-principal), then I is S-w-finite (respectively, S-w-principal) and IDS ∩ D = I : s for some s ∈ S. (2) Let S be the saturation of S. Then D is an S-SM-domain (respectively, S-factorial domain) if and only if D is an S-SM-domain (respectively, S-factorial domain). (3) Let T be a multiplicative subset of D containing S. If D is an S-SM-domain (respectively, S-factorial domain), then D is a T -SM-domain (respectively, T -factorial domain). (4) If S consists of units of D, then D is an S-SM-domain (respectively, S-factorial domain) if and only if D is an SM-domain (respectively, UFD). (5) Let E be an integral domain and let f : D → E be a ring homomorphism such that every nonzero ideal of E is extended from D with respect to f and for every nonzero ideal I of D, f (Iw )E ⊆ (f (I)E)w . If D is an S-SM-domain (respectively, S-factorial domain), then E is an f (S)-SM-domain (respectively, f (S)-factorial domain). (6) If D is an S-SM-domain (respectively, S-factorial domain), then DS is an SMdomain (respectively, UFD). (7) Assume that for every w-finite type ideal J of D, (JDS )w ∩ D = J : s for some s ∈ S. If DS is an SM-domain, then D is an S-SM-domain. Proof. (1) Since IDS ∩D is S-w-finite (respectively, S-w-principal), there exist an s1 ∈ S and a w-finite type (respectively, principal) ideal J of D such that s1 (IDS ∩ D) ⊆ J ⊆ (IDS ∩ D)w = IDS ∩ D, where the equality follows from Lemma 1.2(1). Since I is a w-ideal and J is of w-finite type (respectively, principal), we can find an s2 ∈ S such that s2 J ⊆ I. Hence we have s1 s2 I ⊆ s1 s2 (IDS ∩ D) ⊆ s2 J ⊆ I, which shows that I is S-w-finite (respectively, S-w-principal) and IDS ∩ D = I : s1 s2 . (2) The “only if” part is obvious. For “if” part, let I be a nonzero ideal of D. Since D is an S-SM-domain (respectively, S-factorial domain), there exist a t ∈ S and a w-finite type (respectively, principal) ideal J of D such that tI ⊆ J ⊆ Iw . Let s ∈ S be a multiple

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of t. Then sI ⊆ J ⊆ Iw ; so I is S-w-finite (respectively, S-w-principal). Thus D is an S-SM-domain (respectively, S-factorial domain). (3) This assertion is clear. (4) The result easily follows from the fact that D is an SM-domain (respectively, UFD) if and only if every w-ideal of D is of w-finite type (respectively, principal) (cf. [11, page 284]). (5) Clearly, f (S) is a multiplicative subset of E. Let A be a nonzero ideal of E. Then A = f (I)E for some nonzero ideal I of D. Since D is an S-SM-domain (respectively, S-factorial domain), there exist an s ∈ S and a finitely generated (respectively, principal) ideal F of D such that sI ⊆ Fw ⊆ Iw ; so by the assumption, we obtain       f (s)A = f (s)f (I)E ⊆ f (Fw )E ⊆ f (F )E w ⊆ f (Iw )E w = f (I)E w = Aw , where the second equality comes from the fact that f (Iw )E ⊆ (f (I)E)w if and only if (f (Iw )E)w = (f (I)E)w . Note that f (F )E is a finitely generated (respectively, principal) ideal of E. Hence A is f (S)-w-finite (respectively, f (S)-w-principal), and thus E is an f (S)-SM-domain (respectively, f (S)-factorial domain). (6) The assertion is an immediate consequence of (4), (5) and Lemma 1.2(3). (7) Choose any nonzero ideal I of D. Since DS is an SM-domain, (IDS )w = (JDS )w for some finitely generated subideal J of I; so we have (IDS )w ∩ D = (JDS )w ∩ D = (Jw DS )w ∩ D, where the second equality follows from Lemma 1.2(3). By our assumption, (Jw DS )w ∩ D = Jw : s for some s ∈ S; so we obtain   sI ⊆ s (IDS )w ∩ D ⊆ Jw ⊆ Iw . Hence I is S-w-finite, and thus D is an S-SM-domain. 2 Let P be a prime ideal of an integral domain D and set S := D \ P . Then S is a multiplicative subset of D. We say that D is a P -strong Mori domain (P -SM-domain) (respectively, P -factorial domain (or P -unique factorization domain)) if D is an S-SMdomain (respectively, S-factorial domain). Proposition 1.4. Let D be an integral domain and M a maximal w-ideal of D. Then the following assertions are equivalent. (1) D is an M -SM-domain (respectively, M -factorial domain). (2) DM is a Noetherian domain (respectively, PID) and for every nonzero finitely generated (respectively, principal) ideal J of D, Jw DM ∩D = Jw : s for some s ∈ D\M .

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Proof. (1) ⇒ (2) Let A be a nonzero ideal of DM . Then A = IDM for some nonzero ideal I of D. Since D is an M -SM-domain (respectively, M -factorial domain), there exist an s ∈ D \ M and a finitely generated (respectively, principal) ideal J of D such that sI ⊆ Jw ⊆ Iw . Hence we have IDM = sIDM ⊆ Jw DM ⊆ Iw DM . Note that Bw DM = BDM for any nonzero ideal B of D (cf. [25, Theorem 3.9]) because M is a maximal w-ideal of D. Thus IDM = JDM , which indicates that DM is a Noetherian domain (respectively, PID). The second part follows from Proposition 1.3(1). (2) ⇒ (1) Assume that DM is a Noetherian domain (respectively, PID) and let I be a nonzero ideal of D. Then IDM = JDM for some finitely generated (respectively, principal) subideal J of I. Hence we have I ⊆ IDM ∩ D = Jw DM ∩ D = Jw : s for some s ∈ D \ M , which implies that sI ⊆ Jw ⊆ Iw . Therefore I is (D \ M )-w-finite (respectively, (D \ M )-w-principal), and thus D is an M -SM-domain (respectively, M -factorial domain). 2 2. S-strong Mori domains In this section, D is an integral domain, S is a (not necessarily saturated) multiplicative subset of D and a module means a unitary D-module. We start this section with a relation between an S-Noetherian domain and an S-SM-domain. It is clear that if I is an S-finite ideal of D, then Iw is an S-w-finite ideal of D; so I is S-w-finite. Hence every S-Noetherian domain is an S-SM-domain. If D is a DW-domain, then the concept of an S-SM-domain coincides with that of an S-Noetherian domain, because every S-w-finite ideal is S-finite. (Recall that an integral domain D is a DW-domain (or t-linkative domain) if each nonzero ideal of D is a w-ideal, or equivalently, every nonzero prime ideal of D is a w-ideal [18, Proposition 2.2].) Proposition 2.1. The following assertions hold. (1) An S-SM-domain is a w-locally S-Noetherian domain. (2) A w-locally S-Noetherian domain which has finite w-character is an S-SM-domain. Proof. (1) Let M be a maximal w-ideal of an S-SM-domain D and let I be a nonzero ideal of D. If IDM = DM , then we have nothing to prove; so we assume that IDM is a proper ideal of DM . Since D is an S-SM-domain, there exist an s ∈ S and a finitely generated subideal F of I such that sI ⊆ Fw ⊆ Iw . Therefore we have

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sIDM ⊆ (Fw )DM = F DM ⊆ IDM , and hence IDM is S-finite. Thus DM is an S-Noetherian domain. (2) Assume that D is a w-locally S-Noetherian domain which has finite w-character, and let I be a nonzero ideal of D. If I intersects S, then there is nothing to prove (cf. [3, Proposition 2(a)]); so it suffices to consider the case when I ∩ S = ∅. Choose any 0 = a ∈ I. Then a is contained in only finitely many maximal w-ideals of D, say M1 , . . . , Mn . Fix an i ∈ {1, . . . , n}. Since DMi is S-Noetherian, there exist an si ∈ S and a finitely generated subideal Fi of I such that si IDMi ⊆ Fi DMi . Let s := s1 · · · sn and set C := (a) + F1 + · · · + Fn . Then sIDMi ⊆ CDMi . Let M  be a maximal w-ideal of D such that M  = Mi for all i = 1, . . . , n. Then a is a unit in DM  ; so IDM  = DM  = CDM  . Therefore sIDM ⊆ CDM for all maximal w-ideals M of D. Hence we have  sIw = s





IDM

M ∈w-Max(D)



⊆

sIDM

M ∈w-Max(D)



⊆

CDM

M ∈w-Max(D)

= Cw , where the equalities follow from [10, Proposition 2.8(3)]. Note that C is a finitely generated subideal of I. Hence Iw is S-w-finite, and so I is S-w-finite. Thus D is an S-SM-domain. 2 Proposition 2.2. D is an SM-domain if and only if D is an M -SM-domain for every maximal w-ideal M of D. Proof. The “only if” part is clear. For the converse, let I be a nonzero ideal of D. Since D is an M -SM-domain for every maximal w-ideal M of D, there exist an sM ∈ D \ M and a w-finite type ideal WM of D such that sM I ⊆ WM ⊆ Iw . Let FM be a finitely generated subideal of I such that WM = (FM )w . Note that {sM | M is a maximal w-ideal of D} is not contained in any maximal w-ideal of D; so we can take suitable elements sM1 , . . . , sMn ∈ D such that (sM1 , . . . , sMn )w = D. Therefore we have   Iw = (sM1 , . . . , sMn )I w ⊆ (WM1 + · · · + WMn )w = (FM1 + · · · + FMn )w ⊆ Iw , and hence Iw = (FM1 + · · · + FMn )w . Thus D is an SM-domain. 2 Let D be an integral domain, S a (not necessarily saturated) multiplicative subset of D and M a w-module as a D-module. We say that M is S-w-finite if sM ⊆ F for some s ∈ S and some w-finite type submodule F of M ; and M is an S-strong Mori

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module (S-SM-module) if each w-submodule of M is S-w-finite. It is easy to see that M is S-w-finite if and only if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ Nw ⊆ M ; and a w-finite module (respectively, SM-module) is an S-w-finite module (respectively, S-SM-module). Also, note that for a torsion-free D-module M , MP = (Mw )P for all P ∈ w-Max(D) [25, Theorem 3.9]. We next give the S-invariant of the Cohen type theorem for S-SM-domains. To prove this, we need the following lemma. Lemma 2.3. Let M be an S-w-finite w-module as a D-module and N a w-submodule of M which is maximal among non-S-w-finite w-submodules of M . Then (N :D M ) is a prime w-ideal of D with (N :D M ) ∩ S = ∅. Proof. We first show that if (N :D M ) = (0), then (N :D M ) is a w-ideal. While this follows directly from [23, Lemma 1.2], we include a proof for the sake of completeness. Clearly, (N :D M ) is an ideal of D. Let x ∈ (N :D M )w . Then Jx ⊆ (N :D M ) for some J ∈ GV(D); so JxM ⊆ N . Hence xM = (JxM )w ⊆ Nw = N , which says that x ∈ (N :D M ). Thus (N :D M ) is a w-ideal of D. Next, we prove that (N :D M ) is a prime ideal of D. If (N :D M ) = (0), then there is nothing to prove. Assume that (N :D M ) = (0) and deny the conclusion. Then there exist a, b ∈ D \ (N :D M ) such that ab ∈ (N :D M ); so (N + aM )w is S-w-finite by the maximality of N . Hence we can find an s ∈ S and a w-finite type submodule F1 of M such that s(N + aM )w ⊆ F1 ⊆ (N + aM )w . Now, we may assume that F1 = (n1 + am1 , . . . , nl + aml )w for some n1 , . . . , nl ∈ N and some m1 , . . . , ml ∈ M . Let I = {m ∈ M | am ∈ N }. Then I is a D-submodule of M containing N and bM ; so I is also S-w-finite. Hence there are an element t ∈ S and a w-finite type submodule F2 of M such that tI ⊆ F2 ⊆ Iw . Also, we may write F2 = (q1 , . . . , qk )w for some q1 , . . . , qk ∈ I. Let x ∈ N . Then there exists a GV-ideal J1 of D such that sxJ1 ⊆ (n1 + am1 , . . . , nl + aml ). Write J1 = (d1 , . . . , ds ). Then for each i = 1, . . . , s, sxdi = ui1 (n1 + am1 ) + · · · + uil (nl + aml ) for some ui1 , . . . , uil ∈ D. Hence we have   sxJ1 = ui1 (n1 + am1 ) + · · · + uil (nl + aml ) | i = 1, . . . , s   = (ui1 n1 + · · · + uil nl ) + a(ui1 m1 + · · · + uil ml ) | i = 1, . . . , s . Therefore ({ui1 m1 + · · · + uil ml | i = 1, . . . , s}) ⊆ I. Since tI ⊆ F2 and ({ui1 m1 + · · · + uil ml | i = 1, . . . , s}) is finitely generated, we have   tJ2 {ui1 m1 + · · · + uil ml | i = 1, . . . , s} ⊆ (q1 , . . . , qk ) for some J2 ∈ GV(D). Write J2 = (e1 , . . . , ep ). Then for each j = 1, . . . , p, tej (ui1 m1 + · · · + uil ml ) = vij1 q1 + · · · + vijk qk

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for some vij1 , . . . , vijk ∈ D. Hence we have   tJ2 {ui1 m1 + · · · + uil ml | i = 1, . . . , s}   = {vij1 q1 + · · · + vijk qk | i = 1, . . . , s and j = 1, . . . , p} . Therefore we obtain  (ui1 n1 + · · · + uil nl ) + a(ui1 m1 + · · · + uil ml ) | i = 1, . . . , s tJ2   ⊆ (ui1 n1 + · · · + uil nl ) | i = 1, . . . , s tJ2   + a(ui1 m1 + · · · + uil ml ) | i = 1, . . . , s tJ2

stxJ1 J2 =



⊆ (tn1 , . . . , tnl , aq1 , . . . , aqk ). Since x was arbitrary, stN J1 J2 ⊆ (tn1 , . . . , tnl , aq1 , . . . , aqk ) ⊆ N , and hence stN ⊆ (tn1 , . . . , tnl , aq1 , . . . , aqk )w ⊆ Nw = N. This indicates that N is S-w-finite, which is absurd. Thus (N :D M ) is a prime ideal of D. Finally, we show that (N :D M ) ∩ S = ∅. If s1 ∈ (N :D M ) ∩ S, then s1 M ⊆ N . Since M is S-w-finite, there exist an s2 ∈ S and a finitely generated submodule F of M such that s2 M ⊆ Fw . Therefore we have s1 s2 N ⊆ s1 s2 M ⊆ (s1 F )w ⊆ s1 M ⊆ N, which means that N is S-w-finite. This contradicts the choice of N . Thus (N :D M ) ∩ S = ∅. 2 Theorem 2.4. Let M be an S-w-finite w-module as a D-module. Then M is an S-SMmodule if and only if the D-submodules (P M )w are S-w-finite for all prime w-ideals P of D disjoint from S. Proof. (⇒) This follows from the definition of S-SM-modules. (⇐) Suppose to the contrary that M is not an S-SM-module, and consider the set T = {U | U is a w-submodule of M which is not S-w-finite}. Then T = ∅. Let {Uα }α∈Λ  be a chain in T and set U := α∈Λ Uα . If a ∈ Uw , then there is a GV-ideal J of D such that Ja ⊆ U . Since J is finitely generated, Ja ⊆ Uβ for some Uβ ; so a ∈ (Uβ )w = Uβ ⊆ U . Hence U is a w-submodule of M . If U is S-w-finite, then there exist an s ∈ S and a w-finite type submodule F of M such that sU ⊆ F ⊆ U . Since F is of w-finite type, F ⊆ Uα for some Uα ; so sUα ⊆ F ⊆ Uα . This contradicts the choice of Uα . Hence U is not S-w-finite. Clearly, U is an upper bound of {Uα }α∈Λ ; so by Zorn’s lemma, we can choose a maximal element N in T . By Lemma 2.3, (N :D M ) is a prime (w-)ideal of D.

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Now, we claim that (N :D M ) is disjoint from S. The assertion is clear when (N :D M ) = (0). Assume that (N :D M ) = (0) and (N :D M ) intersects S. Choose any t1 ∈ (N :D M ) ∩ S. Since M is S-w-finite, t2 M ⊆ (b1 D + · · · + bn D)w for some t2 ∈ S and some b1 , . . . , bn ∈ M ; so we have t1 t2 N ⊆ t1 t2 M ⊆ t1 (b1 D + · · · + bn D)w ⊆ N. Therefore N is S-w-finite, a contradiction. Hence (N :D M ) is disjoint from S. Note that   (N :D M ) ⊆ N :D (b1 D + · · · + bn D)w ⊆ (N :D t2 M )   = (N :D M ) : t2 = (N :D M ), where the last equality comes because (N :D M ) is a prime ideal of D disjoint from S. Therefore (N :D M ) = (N :D (b1 D + · · · + bn D)w ). Since N is a w-module, (N :D (b1 D + · · · + bn D)w ) = (N :D b1 D) ∩ · · · ∩ (N :D bn D); so (N :D M ) = (N :D b1 D) ∩ · · · ∩ (N :D bn D). Again, since (N :D M ) is a prime ideal, (N :D M ) = (N :D bi D) for some i ∈ {1, . . . , n}. Clearly, bi ∈ / N . By the maximality of N , (N + bi D)w is S-w-finite; so there exist u1 ∈ S, n1 , . . . , np ∈ N and a1 , . . . , ap ∈ D such that u1 (N + bi D)w ⊆ (n1 + a1 bi , . . . , np + ap bi )w . Let N  = n1 D + · · · + np D. Then by arguing as in the proof of Lemma 2.3, we obtain     u1 N ⊆ N  + (N :D M )bi w ⊆ N  + (N :D M )M w . By the hypothesis, ((N :D M )M )w is S-w-finite; so u2 ((N :D M )M )w ⊆ Gw for some u2 ∈ S and some finitely generated D-submodule G of ((N :D M )M )w . Hence we have     u1 u2 N ⊆ u2 N  + (N :D M )M w ⊆ u2 N  + G w . Since (u2 N  + G)w is a w-finite type submodule of N , N is S-w-finite. However this is absurd. Thus M is an S-SM-module. 2 Note that D itself is an S-w-finite w-module as a D-module; so we have Corollary 2.5. D is an S-SM-domain if and only if every prime w-ideal (disjoint from S) is S-w-finite. Our next work is to study the Hilbert basis theorem for an S-SM-domain when S has a special property. To do this, we need the following lemmas.

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Lemma 2.6. Let M be a nonzero D-module and L be a submodule of M such that M/L is a w-module. If N is a submodule of M containing L, then (N/L)w = Nw /L. Proof. Let a + L ∈ (N/L)w . Then there exists a J ∈ GV(D) such that J(a + L) ⊆ N/L; so Ja ⊆ N . Therefore a ∈ Nw , and hence a + L ∈ Nw /L. Thus (N/L)w ⊆ Nw /L. For the reverse containment, choose any b + L ∈ Nw /L. Then Jb ⊆ N for some GV-ideal J of D; so J(b + L) = Jb + L ⊆ N/L. Hence b + L ∈ (N/L)w , and thus Nw /L ⊆ (N/L)w . 2 Lemma 2.7. Let D be an integral domain and S a multiplicative subset of D. Then the following assertions hold. (1) Let 0 → M  → M → M  → 0 be a short exact sequence of w-modules as D-modules. Then M is an S-SM-module if and only if M  and M  are S-SM-modules.

k (2) Let M1 , . . . , Mk be w-modules. Then i=1 Mi is an S-SM-module if and only if each Mi is an S-SM-module. (3) If D is an S-SM-domain, then every w-finite type torsion-free w-module as a D-module is an S-SM-module. Proof. (1) Let φ : M  → M and ψ : M → M  be D-module homomorphisms in the short exact sequence 0 → M  → M → M  → 0, and set L := Im(φ) = Ker(ψ). (⇒) Assume that M is an S-SM-module. Since M  is isomorphic to a D-submodule of M , M  is an S-SM-module. Let N be a w-submodule of M  . Then N = N0 /L for some D-submodule N0 of M containing L. Note that N0 is a w-module by Lemma 2.6; so N0 is S-w-finite. Therefore we can find an s ∈ S and a finitely generated D-submodule F of N0 such that sN0 ⊆ Fw . Hence we obtain   sN = (sN0 + L)/L ⊆ (Fw + L)/L ⊆ (Fw + L)/L w ⊆ (N0 /L)w = N. Therefore N is S-w-finite, because F/L is finitely generated. Thus M  is an S-SMmodule. (⇐) Let N be a w-submodule of M . Then N/(N ∩ L) is a w-module by Lemma 2.6. Note that N/(N ∩L) ∼ = (N +L)/L ⊆ M/L ∼ = M  ; so N/(N ∩L) is S-w-finite, because M  is an S-SM-module. Hence there exist an s ∈ S and a finitely generated D-submodule H of N containing N ∩ L such that s(N/(N ∩ L)) ⊆ (H/(N ∩ L))w ; so sN ⊆ Hw + (N ∩ L) by Lemma 2.6. Since M  is an S-SM-module, L is also an S-SM-module. Hence (N ∩ L)w is S-w-finite; so we can find a t ∈ S and a finitely generated D-submodule F of (N ∩ L)w such that t(N ∩ L)w ⊆ Fw . Therefore we obtain     stN ⊆ t Hw + (N ∩ L) ⊆ t Hw + (N ∩ L)w ⊆ tHw + Fw ⊆ (tH + F )w . Since tH + F is finitely generated, N is S-w-finite. Thus M is an S-SM-module. (2) This equivalence comes directly from (1).

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(3) Let M be a w-finite type w-module as a D-module, and let F be a finitely generated D-module such that M = Fw . Since F is torsion-free, we can imbed F into a finitely generated free module H k . Therefore M = Fw ⊆ (H k )w = H k . Note that H k is an S-SM-module by (2). Thus M is an S-SM-module by (1). 2 Recall that a multiplicative subset S of an integral domain D is anti-archimedean if n n≥1 s D ∩ S = ∅ for every s ∈ S. For example, if V is a valuation domain with no height-one prime ideals, then V \ {0} is an anti-archimedean subset of V [4, Proposition 2.1]. We are now ready to prove the Hilbert basis theorem for an S-SM-domain when S is anti-archimedean. 

Theorem 2.8. Let D be an integral domain and S an anti-archimedean subset of D. Then D is an S-SM-domain if and only if the polynomial ring D[X1 , . . . , Xn ] is an S-SM-domain. Proof. Note that S is anti-archimedean in every ring containing D as a subring; so it suffices to prove the case n = 1. Set X = X1 . (⇒) Assume that D is an S-SM-domain. Let A be a w-ideal of D[X] and let I be the ideal of D generated by the leading coefficients of the elements in A. Then Iw is an S-w-finite ideal of D; so sIw ⊆ (a1 , . . . , am )w for some s ∈ S and a1 , . . . , am ∈ I. Let f1 , . . . , fm be polynomials in A whose leading coefficients are a1 , . . . , am , respectively, and let n1 , . . . , nm be degrees of f1 , . . . , fm , respectively. Now, let n be the maximum of n1 , . . . , nm , and set B := (f1 , . . . , fm )D[X]. Let f ∈ A with leading coefficient a and degree k. Then sa ∈ (a1 , . . . , am )w ; so Jsa ⊆ (a1 , . . . , am ) for some J ∈ GV(D). Let J = m (d1 , . . . , dt ). Note that for each i = 1, . . . , t, di sa = j=1 rij aj for some ri1 , . . . , rim ∈ D. m If k ≥ n, let gi = di sf − j=1 rij fj X k−nj for each i = 1, . . . , t. Then gi belongs to A and has degree strictly less than k. If we still have deg gi ≥ n for some i ∈ {1, . . . , t}, we repeat the same process. After finitely many steps, we can find a J  ∈ GV(D) and an integer q ≥ 1 such that J  sq f ⊆ (A ∩M ) +B, where M = D⊕DX ⊕· · ·⊕DX n−1 . Since D is an S-SM-domain, Mw is an S-SM-module by Lemma 2.7(3); so (A ∩M )w is S-w-finite. Therefore t(A ∩ M )w ⊆ (Dh1 + · · · + Dhs )w for some t ∈ S and h1 , . . . , hs ∈ A ∩ M . Note that (Dh1 + · · · + Dhs )w ⊆ ((h1 , . . . , hs )D[X])w . (To see this, let h ∈ (Dh1 + · · · + Dhs )w . Then there exists a GV-ideal J  of D such that J  h ⊆ (Dh1 + · · · + Dhs ); so J  D[X]h ⊆ (h1 , . . . , hs )D[X]. Note that J  D[X] ∈ GV(D[X]) [25, Proposition 1.2] (or [9, Lemma 3.1]); so h ∈ ((h1 , . . . , hs )D[X])w .) Let C = (h1 , . . . , hs )D[X] and choose a  u ∈ i≥1 si D ∩ S. Then we obtain   J  D[X]utf ⊆ (A ∩ M ) + B tD[X] ⊆ Cw + B. Note that J  D[X] ∈ GV(D[X]) [25, Proposition 1.2] (or [9, Lemma 3.1]); so utf ∈ (Cw + B)w = (C + B)w . Since the choice of C is independent of that of f , it follows that

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utA ⊆ (C + B)w . Since C + B is finitely generated as an ideal of D[X], A is S-w-finite. Thus D[X] is an S-SM-domain. (⇐) Let I be a nonzero ideal of D. Since D[X] is an S-SM-domain, ID[X] is S-w-finite; so there exist an s ∈ S and a finitely generated subideal J of I such that sID[X] ⊆ (JD[X])w ⊆ (ID[X])w . Note that (AD[X])w = Aw D[X] for all nonzero ideals A of D [8, Proposition 4.3]. Hence sI ⊆ Jw ⊆ Iw , and thus D is an S-SM-domain. 2 Remark 2.9. The S-strong Mori property does not carry over to the power series ring. In fact, there is an example of an SM-domain D such that D[[X]] is not an SM-domain [19, Theorem 10]. (This is the case when S consists of units of D.) For an f ∈ D[X], c(f ) denotes the ideal of D generated by the coefficients of f , and for an ideal J of D[X], c(J) stands for the ideal of D generated by the coefficients of elements in J, i.e., c(J) = f ∈J c(f ). Let Nv = {f ∈ D[X] | c(f )v = D}. Then Nv is a (saturated) multiplicative subset of D[X] and the quotient ring D[X]Nv is called the t-Nagata ring of D. It was shown in [20, Theorem 6.10.5] that D[X]Nv is a DW-domain. Theorem 2.10. If S is an anti-archimedean subset of an integral domain D, then the following statements are equivalent. (1) D is an S-SM-domain. (2) D[X]Nv is an S-SM-domain. (3) D[X]Nv is an S-Noetherian domain. Proof. (1) ⇒ (3) Let ID[X]Nv be a nonzero ideal of D[X]Nv , where I is a nonzero ideal of D[X]. Note that D[X] is an S-SM-domain by Theorem 2.8, because D is an S-SM-domain and S is an anti-archimedean subset of D. Therefore we can find an s ∈ S and a finitely generated subideal J of I such that sI ⊆ Jw ⊆ Iw . Hence we have sID[X]Nv ⊆ Jw D[X]Nv = JD[X]Nv ⊆ ID[X]Nv , the equality follows from Lemma 1.2(3) and the fact that D[X]Nv is a DW-domain. Therefore ID[X]Nv is S-finite, and thus D[X]Nv is an S-Noetherian domain. (3) ⇒ (1) Let I be a nonzero ideal of D. Since D[X]Nv is an S-Noetherian domain, there exist an s ∈ S and a finitely generated subideal J of ID[X] such that sID[X]Nv ⊆ JD[X]Nv ⊆ ID[X]Nv . Let a ∈ I. Then sag ∈ J for some g ∈ Nv ; so sa ∈ c(J)w . Therefore sI ⊆ c(J)w ⊆ Iw , because c(J) ⊆ I. Note that c(J) is finitely generated. Hence I is S-w-finite, and thus D is an S-SM-domain. (2) ⇔ (3) It suffices to note that D[X]Nv is a DW-domain. 2 Theorem 2.11. Let D be an integral domain, S a multiplicative subset of D and M a w-module as a D-module. Then the following assertions hold.

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(1) If L is an S-SM-submodule of M such that M/L is an S-SM-module, then M is an S-SM-module. (2) If D is an S-SM-domain and M is an S-w-finite torsion-free D-module, then M is an S-SM-module. Proof. (1) Let N be a w-submodule of M . If N ⊆ L, then there is nothing to prove, because L is an S-SM-module. Assume that N  L. Note that N/(N ∩ L) is a w-module by Lemma 2.6. Since N/(N ∩ L) ∼ = (L + N )/L ⊆ M/L and M/L is an S-SM-module, N/(N ∩L) is S-w-finite. Hence there exist an s ∈ S and a finitely generated D-submodule H of N containing N ∩ L such that s(N/(N ∩ L)) ⊆ (H/(N ∩ L))w = Hw /(N ∩ L), where the equality follows from Lemma 2.6. Therefore sN ⊆ Hw + (N ∩ L). Also, since L is an S-SM-module, there exist a t ∈ S and a finitely generated D-submodule F of (N ∩ L)w such that t(N ∩ L)w ⊆ Fw . Therefore we have   stN ⊆ t Hw + (N ∩ L) ⊆ tHw + Fw ⊆ (tH + F )w . Note that tH + F is a finitely generated submodule of N . Hence N is S-w-finite, and thus M is an S-SM-module. (2) Let N be a w-submodule of M . Since M is an S-w-finite D-module, there exist an s ∈ S and a finitely generated D-submodule F of M such that sM ⊆ Fw ; so sN ⊆ Fw . Note that Fw is an S-SM-module by Lemma 2.7(3); so we can find a t ∈ S and a finitely generated D-submodule H of sN such that tsN ⊆ Hw . Hence N is S-w-finite, and thus M is an S-SM-module. 2 Let M be a unitary D-module. We say that M is w-faithfully flat if M is w-flat and (M/P M )w = 0 for all maximal w-ideals P of D. (Recall that M is w-flat if MP is flat for all maximal w-ideals P of D.) It was shown that M is w-faithfully flat if and only if M is w-locally faithfully flat, i.e., the DP -module MP is faithfully flat for all maximal w-ideals P of D [14, Proposition 2.5]; and that for an extension D ⊆ E of integral domains, if E is a w-faithfully flat D-module, then (IE)w ∩ D = Iw for any nonzero ideal I of D [14, Lemma 2.10]. The following result shows that the S-strong Mori property descends into a w-faithfully flat ring extension. Proposition 2.12. Let D ⊆ E be an extension of integral domains such that (IE)w ∩ D = Iw for each nonzero ideal I of D, and let S be a multiplicative subset of D. If E is an S-SM-domain, then so is D. Proof. Let I be a nonzero ideal of D. Since E is an S-SM-domain, there exist an s ∈ S and g1 , . . . , gn ∈ I such that s(IE) ⊆ ((g1 , . . . , gn )E)w . Therefore by the assumption, we have

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    sI ⊆ (sI)w = (sIE)w ∩ D ⊆ (g1 , . . . , gn )E w ∩ D = (g1 , . . . , gn )D w ⊆ Iw , and hence I is S-w-finite. Thus D is an S-SM-domain. 2 3. S-factorial domains Throughout this section, D always denotes an integral domain and S is a (not necessarily saturated) multiplicative subset of D. Our first theorem is a generalization of the Cohen type theorem due to Isaacs. To do this, we need the following lemma. Lemma 3.1. A w-ideal M of D maximal among non-S-w-principal ideals is a prime ideal of D. Proof. Let M be a w-ideal of D maximal among non-S-w-principal ideals of D, and suppose to the contrary that M is not a prime ideal. Then we can find suitable elements a, b ∈ D \ M such that ab ∈ M ; so M + (a) is S-w-principal by the maximality of M . Hence there exist an s ∈ S and a principal ideal (c) of D such that s(M + (a)) ⊆ (c) ⊆ (M + (a))w . Let I := {d ∈ D | dc ∈ M }. Then I is an ideal of D containing M and b; so I is also S-w-principal by the maximality of M . Therefore we can find a t ∈ S and an ideal (e) of D such that tI ⊆ (e) ⊆ Iw . Let x ∈ M . Then sx = cy for some y ∈ I; so sM ⊆ Ic ⊆ M . Hence we have stM ⊆ tIc ⊆ (ce) ⊆ (Ic)w ⊆ Mw = M, which means that M is S-w-principal. However, this is absurd. Thus M is a prime ideal of D. 2 Note again that a nonzero ideal I is S-w-principal if and only if Iw is S-principal (Remark 1.1(2)). Theorem 3.2. D is an S-factorial domain if and only if every prime w-ideal of D (disjoint from S) is S(-w)-principal. Proof. (⇒) This implication is clear. (⇐) Suppose that D is not an S-factorial domain, and let Ω be the set of non-S(-w)-principal w-ideals of D. Then Ω is nonempty and is partially ordered un der inclusion. Let {Pα }α∈Λ be a chain in Ω and set P := α∈Λ Pα . If a ∈ Pw , then Ja ⊆ P for some J ∈ GV(D). Since J is finitely generated, Ja ⊆ Pα for some Pα ; so a ∈ (Pα )w = Pα ⊆ P . Hence P is a w-ideal of D. If P is S(-w)-principal, then there exist an s ∈ S and a principal ideal (d) of D such that sP ⊆ (d) ⊆ P ; so (d) ⊆ Pβ for some Pβ . Therefore sPβ ⊆ (d) ⊆ Pβ . This is impossible, because Pβ is not S(-w)-principal. Hence P is not S(-w)-principal. Note that P is an upper bound of {Pα }α∈Λ ; so by Zorn’s

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lemma, there exists a maximal element M in Ω. By Lemma 3.1, M is a prime ideal, which contradicts the hypothesis. Thus D is an S-factorial domain. 2 Let D be an integral domain and S a multiplicative subset of D. Recall that D is a Prüfer v-multiplication domain (PvMD) if every nonzero finitely generated ideal of D is w-invertible (or equivalently, DP is a valuation domain for all prime w-ideals P of D [7]); and D is well behaved if for each prime t-ideal P of D, P DP is a prime t-ideal of DP . (Note that P DP is not generally a t-ideal of DP for a prime t-ideal P of D [26, Proposition 2.5].) It was shown in [26, Corollary 1.3] that if P is a prime t-ideal of a well behaved domain D such that P ∩ S = ∅, then P DS is a prime t-ideal of DS . Note that w = t in a PvMD; and that PvMDs are well behaved [26, page 202]. Hence if P is a prime w-ideal of a PvMD D such that P ∩ S = ∅, then P DS is also a prime w-ideal of DS . Corollary 3.3. Let P be a subset of a PvMD D consisting of representatives for the principal primes of DS . If DS is a UFD, then D is an S-factorial domain if and only if pDS ∩ D is S-w-finite for each p ∈ P. Proof. The necessary condition is clear. For the converse, let P be a prime w-ideal of D. If P ∩ S = ∅, then P is S-principal [3, Proposition 2(a)]; so we assume that P is disjoint from S. Since D is a PvMD, P DS is a prime w-ideal of DS . Since DS is a UFD, P DS = pDS for some p ∈ P (cf. [11, page 284]). Note that P = P DS ∩ D = pDS ∩ D. By the hypothesis, there exist an s ∈ S and a w-finite type ideal F of D such that sP ⊆ F ⊆ P . Since F is of w-finite type, stP ⊆ tF ⊆ (p) ⊆ P for some t ∈ S. Hence P is S(-w)-principal. Thus by Theorem 3.2, D is an S-factorial domain. 2 Let D be an integral domain and S a (saturated) multiplicative subset of D. We say that S is a t-splitting set of D if for each 0 = d ∈ D, (d) = (AB)t for some integral ideals A, B of D such that At ∩ (s) = sAt (or equivalently, (A, s)t = D (cf. [1, Lemma 2.4])) for all s ∈ S and Bt ∩ S = ∅. It was shown in [1, Corollary 2.3] that S is a t-splitting set of D if and only if for all 0 = d ∈ D, dDS ∩ D is w-invertible in D; so if S is a t-splitting set of D, then for all 0 = d ∈ D, dDS ∩ D is of w-finite type [10, Proposition 2.6], and hence S-w-finite. Thus by Corollary 3.3, we have Corollary 3.4. Let D be a PvMD such that DS is a UFD. If S is a t-splitting set of D, then D is an S-factorial domain. Recall that D is a Krull domain if there exists a family {Vα }α∈Λ of rank-one essential  discrete valuation overrings of D such that D = α∈Λ Vα and this intersection has finite character, i.e., each nonzero nonunit in D is a nonunit in only finitely many of valuation overrings Vα . It was shown in [24, Theorem 2.8] that D is a Krull domain if and only if D is both a PvMD and an SM-domain. We also note that an SM-domain is always an S-SM-domain; and every multiplicative subset of a Krull domain is t-splitting [1, page 8].

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Hence as immediate consequences of Proposition 1.3(6) and Corollaries 3.3 and 3.4, we obtain Corollary 3.5. The following assertions hold. (1) If D is both a PvMD and an S-SM-domain, then DS is a UFD if and only if D is an S-factorial domain. (2) If D is a Krull domain, then D is an S-factorial domain if and only if DS is a UFD. Note that D is a Krull domain if and only if every nonzero ideal of D is w-invertible (cf. [11, Theorem 3.6]); so D is a Krull domain if and only if D is an SM-domain and DM is a PID for all maximal w-ideals M of D (cf. [10, Proposition 2.6]). We are closing this article with new characterizations of Krull domains. Proposition 3.6. D is a Krull domain if and only if D is an M -factorial domain for all maximal w-ideals M of D. Proof. (⇒) Assume that D is a Krull domain, and let M be a maximal w-ideal of D. Then D is an SM-domain; so by Proposition 2.2, D is an M -SM-domain. Note that DM is a PID. Thus by Proposition 1.4, D is an M -factorial domain. (⇐) Clearly, any M -factorial domain is an M -SM-domain; so D is an M -SM-domain for all maximal w-ideals M of D. Hence D is an SM-domain by Proposition 2.2. Also, DM is a PID by Proposition 1.4. Thus D is a Krull domain. 2 For a prime w-ideal P of an integral domain D, the w-height of P , denoted by w-ht(P ), is the supremum of lengths of chains of prime w-ideals between (0) and P . (Here, we regard (0) as a prime w-ideal of D.) The w-dimension of D, denoted by w-dim(D), is defined to be the supremum of {w- ht(P ) | P is a prime w-ideal of D}. Note that if D is a Krull domain, then w-dim(D) = 1, because in a Krull domain, every w-ideal is a v-ideal and every height-one prime ideal is a maximal v-ideal. Corollary 3.7. D is a Krull domain if and only if for all prime w-ideals P ⊆ Q of D, bP ⊆ (a) for some a ∈ P and b ∈ D \ Q. Proof. (⇒) Let P ⊆ Q be prime w-ideals of D. Since D is a Krull domain, w-dim(D) = 1; so P = Q and P is a maximal w-ideal of D. Again, since D is a Krull domain, by Proposition 3.6, D is a P -factorial domain; so the result holds. (⇐) Let Q be any maximal w-ideal of D, set S := D \Q, and choose any prime w-ideal P of D. If P ∩ S = ∅, then P is S-principal [3, Proposition 2(a)]; so P is S-w-principal. If P ∩ S = ∅, then our assumption forces P to be S-w-principal; so D is a Q-factorial domain by Theorem 3.2. Thus D is a Krull domain by Proposition 3.6. 2

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