Houston Journal of Mathematicsc© 2008 University of Houston

Volume 34, No. 2, 2008

INJECTIVE MODULES OVER STRONG MORI DOMAINS

HWANKOO KIM], EUN SUP KIM, AND YOUNG SOO PARK

Communicated by Johnny A. Johnson

Abstract. We show that an integral domain R is a strong Mori domain

if and only if any direct sum of co-semi-divisorial injective R-modules is

injective. We also show that if M is a nonzero co-semi-divisorial torsion

module over a strong Mori domain R with w-dim(R) = 1, then W (M) ∼=⊕P∈w-Max(R)

MP , where W (M) is the semi-divisorial envelope of M .

1. Introduction

Let R be an integral domain. In [10], H. Kim extended to any R-module thenotion of semi-divisorial closure, or w-closure, defined by Wang and McCaslandfor torsion-free R-modules in [17], and introduced and studied the related notionsof co-semi-divisoriality and w-nullity. These concepts were then used to givenew module-theoretic characterizations of t-linkative domains, a class of domainswidely considered in multiplicative ideal theory.

On the other hand, in [2, 13, 14, 16], the authors investigated injective (resp.,flat) modules over a Krull domain in terms of co-divisorial modules, pseudo-nullmodules, and divisorial modules. In particular, it was shown that for a completelyintegrally closed domain R, R is a Krull domain if and only if any direct sum ofcodivisorial injective R-modules is injective ([2, Proposition 2.7] or [13, Remark7]), which is an analogue of the well-known result that a ring R is Noetherian ifand only if any direct sum of injective R-modules is injective. Recently, in [12]Moucouf investigated injective modules over a ring of Krull type and obtained a

2000 Mathematics Subject Classification. 13A15, 13Bxx, 13Cxx, 13D07, 13Dxx, and 13G05.Key words and phrases. Injective module, co-semi-divisorial, strong Mori domain, semi-

divisorial (envelope).] Corresponding Author

349

350 H. KIM, E. S. KIM, AND Y. S. PARK

generalization of ([2, Proposition 2.7]). Also in [5], L. Fuchs investigated injectivemodules over the special case of Mori domains. Among other things, L. Fuchsobtained the following two results, which are strictly related to this paper. (Forunexplained terminologies and notations, we refer to [5])

Theorem 1.1. ([5, Corollary 8]) Every injective module E over a P-NoetherianMori domain R is a direct sum E = D⊕C, such that D is a direct sum of injectivesof the form E(R/P ), where the P are divisorial primes with RP Noetherian, andthe elements of C have divisorial-free annihilator ideals. This decomposition isunique up to isomorphism.

Theorem 1.2. ([5, Theorem 9]) For a domain R with quotient field K, thefollowing conditions are equivalent:

(i) the module E(K/R) is Σ-injective;(ii) R is a P-Noetherian Mori domain;(iii) R is a Mori domain such that RP is Noetherian for every P ∈ P;(iv) R is a strong Mori domain.

In Section 2, we investigate injective modules over strong Mori domains usinga different approach with respect to Fuchs and obtain a generalization of ([2,Proposition 2.7] or [13, Remark 7]). That is, an integral domain R is a strongMori domain if and only if any direct sum of co-semi-divisorial injective R-modulesis injective. In Section 3, we also show that if M is a nonzero co-semi-divisorialtorsion module over an SM domain R with w-dim(R) = 1, then the semi-divisorialenvelope W (M) ∼=

⊕P∈w-Max(R)

MP .

Throughout this paper, R denotes an integral domain with quotient field K. LetF(R) (resp., f(R)) denote the set of nonzero (resp., nonzero finitely generated)fractional ideals of R. A ∗-operation (star operation) on R is a mapping A→ A∗from F(R) to F(R) which satisfies the following conditions for all a ∈ K − {0}and A,B ∈ F(R): (1) (a)∗ = (a) and (aA)∗ = aA∗, (2) A ⊆ A∗; if A ⊆ B, thenA∗ ⊆ B∗, and (3) (A∗)∗ = A∗.

For details on star operations, the reader may consult [7, sections 32 and 34].An A ∈ F(R) is called a ∗-ideal if A∗ = A. Recall that the function on F(R)defined by A → (A−1)−1 = Av is a star operation called the v-operation. Thet-operation on R is the star operation defined by A → At = ∪{Jv|J ⊆ A withJ ∈ F(R) finitely generated}. The identity mapping on F(R) is obviously astar operation; it is called the d-operation. For any star operation ∗ and for anyA ∈ F(R), we have A ⊆ A∗ ⊆ Av, and hence (A∗)v = Av. In particular, a v-ideal(or divisorial ideal) is a ∗-ideal for any ∗.

INJECTIVE MODULES OVER SM DOMAINS 351

An ideal J of R is called a Glaz-Vasconcelos ideal (for short, GV-ideal, denotedby J ∈ GV(R)) if J is a finitely generated ideal of R with J−1 = R. Following[17], a torsion-free R module M is called a w-module if Jx ⊆ M for J ∈ GV(R)and x ∈ M ⊗K imply that x ∈ M , which is said to be semi-divisorial in [8] andan F∞-ideal in the ideal case in [9]. For a torsion-free R-module M , Wang andMcCasland defined the w-envelope of M in [17] as Mw = {x ∈ M ⊗ K | Jx ⊆M for some J ∈ GV(R)}. In particular, if I is a nonzero fractional ideal, thenIw = {x ∈ K | Jx ⊆ R for some J ∈ GV(R)}. The canonical map I 7→ Iwon F(R) is a star-operation, denoted w. We shall consider the zero ideal in R

as a w-ideal. In [17, 18], Wang and McCasland defined an integral domain R

to be a strong Mori domain (for short, SM domain) if R satisfies the ACC onw-ideals. It was shown in [17] that a prime ideal P of R is a w-ideal if and onlyif Pw 6= R. Therefore, all prime ideals contained in a proper w-ideal of R arealso w-ideals. Thus they define w-dim(R) = sup{ht(P ) | P ∈ w-Max(R)}, wherew-Max(R) is the set of w-maximal ideals of R. From the definition we see thatthe w-dimension is also the supremum of the lengths of chains of w-primes. It iswell-known that Krull domains have w-dim(R) ≤ 1. If dim(R) = 1, it is clear thatw-dim(R) = 1. It is also worth noting that w distributes over (finite) intersections[17, Proposition 2.5]. For unexplained terminologies and notations, we refer to[1, 7, 15].

The methods used in this paper are somewhat similar to those found in [2] and[13].

2. Co-semi-divisorial injective modules

In [10], H. Kim introduced the notions of “co-semi-divisoriality” and “w-nullity” of a module as follows. Let M be a module over an integral domainR and let M := {x ∈ M | (O(x))w = R}, where O(x) := (0 :R x) = annR(x)is the order ideal of x. Then M is a submodule of M . M is said to be co-semi-divisorial (resp., w-null) if M = 0 (resp., M = M). Note that the notions ofco-semi-divisoriality and w-nullity can be interpreted in terms of a suitable tor-sion theory [6, IX, Proposition 6.2 and Proposition 6.4] (with P = w-Max(R)).The following result will be useful in the sequel.

Proposition 2.1. ([10, Proposition 2.6]) The following statements are equivalentfor an R-module M .

(1) M is co-semi-divisorial.(2) O(x) is a w-ideal for each element x ∈M .(3) (O(x))w 6= R for each nonzero element x ∈M .

352 H. KIM, E. S. KIM, AND Y. S. PARK

(4) HomR(N,M) = 0 for each w-null R-module N .

Note from [17, Proposition 1.4] that the annihilator ideal of any submodule ofa co-semi-divisorial module is a w-ideal.

Recall from [2] that a module M is said to be codivisorial if the annihilatorof every nonzero element of M is a divisorial ideal. Thus in a Krull domain, thenotions of co-semi-divisoriality and codivisoriality are the same.

For anR-moduleM , we letAssR(M) = {P ∈ Spec(R) | P = O(x) for some x ∈M}. Note that AssR(M) = {P ∈ Spec(R) | R/P has an injection into M}. Asusual, ER(M) denotes the injective envelope (or injective hull) of M . If no con-fusion arises, we write Ass(M) (resp., E(M)) for AssR(M) (resp., ER(M)). It iseasy to see that Ass(M) = Ass(E(M)) for any R-module M .

The following proposition is due to I. Beck [2, Proposition 2.1]; but we includea proof for the sake of completeness. Note that [2, Proposition 2.1] holds withoutthe assumption that the prime ideal P is of height one.

Proposition 2.2. Let P be a prime ideal of an integral domain R and let f :I → E(R/P ) be a homomorphism. Suppose that f(x) = 0 for some x ∈ I \ P .Then f ≡ 0.

Proof. Since E(R/P ) is injective, we can extend the R-homomorphism f to ahomomorphism f : R → E(R/P ). Let J = ker(f). If J = R, there is nothing toprove. Otherwise, E(R/P ) ∼= E(R/J) since E(R/P ) is indecomposable by [11,Theorem 2.4] and there is an injection R/J → E(R/P ). Hence P ∈ Ass(R/P ) =Ass(E(R/P )) = Ass(E(R/J)) = Ass(R/J), and so J ⊆ P . This contradictsf(x) = 0, hence J = R. �

Recall from [17, Theorem 4.9] that if I is a proper w-ideal of an SM domain R,then there are only finitely many prime ideals minimal to I, and every minimalprime of I is of the form I :R r for some r ∈ R \ I.

Proposition 2.3. Let R be an SM domain with quotient field K. Let M be anonzero co-semi-divisorial indecomposable injective R-module. Then M ∼= K orM ∼= E(R/P ) for some w-prime ideal of R.

Proof. If M is not a torsion module, then we have that M ∼= K. Hence we mayassume that M is a torsion module. Let 0 6= x ∈M and let I = O(x) (Note thatI R). We have an injection R/I → M and since M is indecomposable, andR/I 6= 0, it follows that M ∼= E(R/I). Since M is co-semi-divisorial, I is a w-idealby Proposition 2.1. By [17, Theorem 4.9] every minimal prime, say P , of I is ofthe form I :R r for some r ∈ R \ I. (Note that P = I :R r is also a w-ideal of R.)

INJECTIVE MODULES OVER SM DOMAINS 353

Hence we have an injection R/Pr·−→ R/I → M , where r· is an r-multiplication

map. Since M is indecomposable, we have that M ∼= E(R/P ). �

The notion of “Σ-injectivity” was introduced in [4] by C. Faith and Σ-injectivemodules were investigated thoroughly in [3] by I. Beck. An injective module isΣ-injective in case an infinite direct sum of copies is injective. It was also shownin [5, Theorem 9] that an integral domain R is an SM-domain if and only if themodule E(K/R) is Σ-injective.

Proposition 2.4. Let P be a prime w-ideal of an SM domain R. Then themodule E(R/P ) is Σ-injective, that is, any direct sum

⊕E(R/P ) is injective.

Proof. Note first that R/P is co-semi-divisorial since P is a w-ideal. Thus by[10, Corollary 2.11] E(R/P ) is also co-semi-divisorial. By Proposition 2.1 O(x)is a w-ideal for each element x ∈ R/P . Thanks to C. Faith [3, Theorem 1.8],an injective module E is Σ-injective if and only if R satisfies the ACC on theannihilators of the submodules of E. Since R is an SM domain, the moduleE(R/P ) is Σ-injective. �

Proposition 2.5. Let R be an SM domain and let E be a co-semi-divisorialR-module which is a direct sum of indecomposable injective modules. Then E isinjective.

Proof. It is enough to consider the case of E torsion. Let f : I → E. Wewill show that we can find an injective submodule E′ ⊆ E such that f(I) ⊆ E′,whence f can be extended to a map f : R→ E, and consequently E is injective.

Let 0 6= x ∈ I. Then x is contained in just a finite number of w-maximal idealsof R, since R is an SM domain. Furthermore, f(x)′s components in the directsum are zero almost everywhere. If x 6∈ P ∈ w-Max(R), g : I → E(R/P ), andg(x) = 0, then by Proposition 2.2 g(I) = {0}. The rest of proof follows fromProposition 2.3 and Proposition 2.4. �

Recall from [17, Definition 5] that a proper w-submodule N of a w-module Mis said to be w-irreducible in M if N is not the intersection of two w-submodulesof M which properly contain N .

Proposition 2.6. Let E be a co-semi-divisorial injective module over an SMdomain R. Then E is a direct sum of indecomposable injective modules.

Proof. Let E be a co-semi-divisorial injective module. Let C be a maximalsubmodule of E with respect to being a direct sum of indecomposable injectivesubmodules. (Note that, in view of Proposition 2.5, the family of submodules of E

354 H. KIM, E. S. KIM, AND Y. S. PARK

which are direct sums of indecomposable injectives is inductive, hence a maximalC exists.) Then by Proposition 2.5, C is injective, and hence E ∼= C⊕D for somesubmodule D of E. Let 0 6= x ∈ D. Then

O(x) = {0} or O(x) = J1 ∩ J2 ∩ · · · ∩ Jk

for some w-irreducible ideal Ji since E is co-semi-divisorial. By [17, Theorem4.10] each Ji is Pi-primary for some prime w-ideal Pi. In the first case, x iscontained in a summand of D isomorphic to K. In the second, we may assumethat this decomposition is irredundant. Note that if J is a w-irreducible ideal ofR, then the module E(R/J) is indecomposable. Indeed, if A and B are ideals ofR contain J such that A/J ∩ B/J = 0, then A ∩ B = J, and so J = Aw ∩ Bw,and thus either J = Aw or J = Bw. Hence either J = A or J = B. ThereforeE(R/J) is indecomposable by [11, Proposition 2.2]. Thus by [11, Theorem 2.3]we have

E(R/O(x)) ∼= E(R/J1)⊕ E(R/J2)⊕ · · · ⊕ E(R/Jk).

Note that if J is a w-irreducible P -primary ideal of R, then E(R/J) ∼= E(R/P ).Indeed, if J = P , we are finished. Hence we assume that J ( P . Then thereis a smallest integer n > 1 such that Pn ⊆ J since R is an SM domain. Takeb ∈ Pn−1 \ J and denote the image of b in R/J by b. Then it is easy to see thatP = O(b). Therefore, there is an element of E(R/J) with the order ideal P , andthus E(R/J) ∼= E(R/P ) by [11, Theorem 2.4]. Hence, we have

E(Rx) ∼= E(R/O(x)) ∼= E(R/P1)⊕ E(R/P2)⊕ · · · ⊕ E(R/Pk).

Both cases contradict the maximality of C, and hence C = E. �

Immediately from the propositions above we have the following two corollaries.

Corollary 2.7. Let R be an SM domain with quotient field K and let M be anonzero co-semi-divisorial injective R-module. Then M is indecomposable if andonly if it is isomorphic to K or E(R/P ) for some w-maximal ideal of R.

Corollary 2.8. Let R be an SM domain and let M be a nonzero co-semi-divisorialR-module. Then M is injective if and only if it is isomorphic to a direct sum ofindecomposable co-semi-divisorial injective modules.

We can now prove the main result of this section.

Theorem 2.9. Let R be an integral domain. Then R is an SM domain if andonly if any direct sum of co-semi-divisorial injective R-modules is injective.

INJECTIVE MODULES OVER SM DOMAINS 355

Proof. The necessity follows from Proposition 2.5 and Proposition 2.6.To prove the sufficiency, we show that if R is not an SM domain, then there is

an ideal I and an R-homomorphism from I to a direct sum of co-semi-divisorialinjective R-modules that cannot be extended to R.

If R is not an SM domain, then there is a strictly increasing sequence of w-ideals I1 ( I2 ( · · · . Let I =

⋃∞n=1 In. Then I is a w-ideal and I/In 6= 0

for all n. Since each In is a w-ideal, I/In is co-semi-divisorial for each n. By[10, Corollary 2.11] we can imbed I/In in a co-semi-divisorial injective R-moduleEn(:= E(I/In)). We claim that

⊕En is not injective. Now the assertion follows

from the technique due to H. Bass who gave the well-known criterion for a ringto be Noetherian in terms of injective modules (for a proof see [15, Theorem4.10]). �

Note that an integral domainR is a Krull domain if and only ifR is an integrallyclosed SM domain [18, Theorem 2.8]. In a Krull domain, the v-operation and thew-operation are the same thing. Thus Theorem 2.9 applied to Krull domainsreduces to [2, Proposition 2.7].

3. Co-semi-divisorial torsion modules

It was shown in [2, Proposition 1.9] and [13, Theorem 4 (ii)] that if M is acodivisorial torsion module over a Krull domain R, then D(M) ∼=

⊕P∈Ass(M)

MP ,

where D(M) is the divisorial envelope of M in the sense of [13].In this section, we generalize this result as follows (Theorem 3.12). Let R be

an SM domain with w-dim(R) = 1. If M is a nonzero co-semi-divisorial torsionR-module, then W (M) ∼=

⊕P∈w-Max(R)

MP , where W (M) is the semi-divisorial

envelope of M .Throughout this section, S will be a fixed multiplicatively closed subset of an

integral domain R.

Proposition 3.1. Let R be an integral domain. If M is a co-semi-divisorialR-module, then MS is a co-semi-divisorial RS-module.

Proof. The assertion follows immediately from Proposition 2.1. �

Corollary 3.2. Let M be a module over an integral domain R. Then (M)S =MS. In particular, (M/M)S ∼= MS/MS.

Proof. Consider the following exact sequence

0→ MS →MS → (M/M)S → 0.

356 H. KIM, E. S. KIM, AND Y. S. PARK

Thus we have MS ⊆ MS by [10, Proposition 2.8] and Proposition 3.1. Conversely,it can be easily seen that MS ⊆ MS by the fact that (IS)w = RS for any ideal Iof R such that Iw = R. �

The following definitions are taken from [10, Definition 6.1]. Let N be an R-module and M be a submodule of N . M is said to be semi-divisorial in N if N/Mis co-semi-divisorial. In particular, M is said to be semi-divisorial if it is semi-divisorial in its injective envelope E(M). Put WR(M ;N) := p−1(N/M), wherep : N → N/M is the canonical homomorphism. Then we say that WR(M ;N) isthe semi-divisorial envelope of M in N . In particular, we denote WR(M ;E(M))by WR(M) and it is called a semi-divisorial envelope of M . If no confusion arises,we write W (M ;N) (resp., W (M)) for WR(M ;N) (resp., WR(M)). It was shownin [10, Corollary 6.3] that M is semi-divisorial if and only if W (M) = M . It wasalso shown in [10, Theorem 9.9] that in the torsion-free case, the semi-divisorialenvelope and the w-envelope are the same thing, and hence the class of semi-divisorial modules is equal to the class of w-modules.

Corollary 3.3. Let N be a module over an integral domain R and M be a sub-module of N . Then WR(M ;N)S = WRS

(MS ;NS).

Proof. The assertion follows immediately from Corollary 3.2. �

Lemma 3.4. Let M be a co-semi-divisorial module over an SM domain R andlet N be an essential extension of M . Then NS is an essential extension of MS.

Proof. Since N is co-semi-divisorial by [10, Proposition 2.9], NS is a co-semi-divisorial RS-module by Proposition 3.1. We may assume that NS 6= 0. Let0 6= x ∈ NS and put x = y/s, where y ∈ N and s ∈ S. Set F = {P ∈AssR(Ry) | P∩S = ∅}. Then AssRS

(xRS) = {PS | P ∈ F}. Thus AssRS(xRS) 6=

∅. If not, xRS = 0, which is a contradiction. Hence F 6= ∅. Let P ∈ F and taker ∈ R such that P = O(ry). By assumption, M∩Rry 6= 0. Take 0 6= z ∈M∩Rryand put z = bry for some b ∈ R. Then 0 6= z/1 ∈MS . If otherwise, there is t ∈ Ssuch that tz = 0, and hence tbry = 0, i.e., tb ∈ O(ry) = P . Since P ∩ S = ∅,b ∈ P . In other words, z = bry = 0. This contradicts the choice of z. HencexRS ∩MS 6= 0. This implies that NS is an essential extension of MS . �

Corollary 3.5. Let M be a co-semi-divisorial module over an SM domain R.Then ER(M)S ∼= ERS

(MS).

Proof. ER(M)S is an essential extension of MS by Lemma 3.4. Hence it is suffi-cient to show that ER(M)S is injective. This follows immediately from Corollaries2.7 and 2.8 and Theorem 2.9. �

INJECTIVE MODULES OVER SM DOMAINS 357

Corollary 3.6. Let M be a co-semi-divisorial module over an SM domain R.Then WR(M)S ∼= WRS

(MS).

Proof. The assertion follows immediately from Corollary 3.3 and Corollary 3.5.�

Corollary 3.7. Let R be an SM domain. Then if M is a co-semi-divisorial andsemi-divisorial R-module, then MS is a co-semi-divisorial and semi-divisorial RS-module.

Proof. The assertion follows from [10, Corollary 5.3] and Corollary 3.6. �

Corollary 3.8. Let R be an SM domain, Λ be a set, and Mλ be a co-semi-divisorial R-module for any λ ∈ Λ. Then we have E(

⊕λMλ) ∼=

⊕λE(Mλ) and

W (⊕

λMλ) ∼=⊕

λW (Mλ).

Proof. It can be easily seen that⊕

λE(Mλ) is an essential extension of⊕

λMλ.Since each E(Mλ) is co-semi-divisorial injective by [10, Corollary 2.11],

⊕λE(Mλ)

is injective by Theorem 2.9, and hence E(⊕

λMλ) ∼=⊕

λE(Mλ) and W (⊕

λ Mλ) ∼=⊕λ W (Mλ). The last assertion follows immediately from [10, Corollary 6.5]. �

In [5], it was shown that an integral domain R (with quotient field K) is anSM domain if and only if the module E(K/R) is Σ-injective. The necessity ofthis fact is an immediate consequence of Corollary 3.8.

Proposition 3.9. Let R be an SM domain and let {Mλ, fλ,µ}λ∈Λ be a directsystem of co-semi-divisorial R-modules and {M,fλ} its inductive limit. Then M

is co-semi-divisorial.

Proof. The assertion follows from a similar method as in the proof of [13, Propo-sition 29]. �

Question 1. In view of Proposition 3.9, if R is an SM domain, is every directlimit (directed indexed set) of co-semi-divisorial injective modules injective?

Recall some definitions from [10]: Let f be a homomorphism of modules over anintegral domain R. Then f is said to be w-injective (resp., w-surjective) if ker(f)(resp., coker(f)) is w-null. And f is said to be w-isomorphic if it is w-injectiveand w-surjective. Furthermore, a monomorphism f is said to be essentially w-isomorphic if it is an essential extension and is also w-isomorphic.

Let M be an R-module and let FM = {(N, f) | N is an R-module and f :M → N is essentially w-isomorphic}. Let (L, g), (N, f) ∈ FM . Then we saythat (L, g) is equivalent to (N, f), denoted by (L, g) = (N, f), if there is an

358 H. KIM, E. S. KIM, AND Y. S. PARK

isomorphism h : L → N such that f = gh. We say that (N, f) is larger than(L, g), denoted by (N, f) > (L, g), if there is a homomorphism j : L → N suchthat f = gj. Furthermore, we say that (L, g) is w-equivalent to (N, f), denoted by(L, g) ∼ (N, f), if (L, g) > (N, f) and (L, g) < (N, f). An element of FM is said tobe an essentially w-isomorphic extension of M . Let (N, f) ∈ FM . (N, f) is calleda maximal essentially w-isomorphic extension (resp., a w-maximal essentially w-isomorphic extension) of M ; simply maximal (resp., w-maximal) in FM if thereis no element (L, g) ∈ FM such that (L, g) > (N, f) and (L, g) is not equivalent(resp., w-equivalent) to (N, f).

Proposition 3.10. ([10, Proposition 7.3]) Let M be an R-module and (N, f) bean element of FM . Then the following conditions are equivalent:

(i) (N, f) is w-maximal in FM .(ii) (N, f) is w-equivalent to (W (M), i), where i : M →W (M) is the canon-

ical injection.(iii) (N, f) is maximal in FM .(iv) (N, f) is equivalent to (W (M), i).

It is shown in [10, Proposition 9.3] that an R-module M is w-null if and onlyif MP = 0 for any w-maximal ideal P of R.

Lemma 3.11. Let f be a homomorphism. Then f is w-injective (resp., w-surjective, w-isomorphic) if and only if fP is injective (resp., surjective, isomor-phic) for any w-maximal ideal P of R.

Proof. The assertion follows immediately from [10, Proposition 9.3]. �

Recall that a module is said to be coirreducible if any two nonzero submoduleshave nonzero intersection.

We can now prove the main result of this section.

Theorem 3.12. Let R be an SM domain with w-dim(R) = 1. If M is a nonzeroco-semi-divisorial torsion R-module, then W (M) ∼=

⊕P∈w-Max(R)

MP . In particu-

lar, if M is a nonzero coirreducible co-semi-divisorial torsion module over an SMdomain with w-dim(R) = 1, then W (M) ∼= MP , where AssR(M) = {P}.

Proof. For each P ∈ w-Max(R), let iP be the canonical homomorphism of M toMP and let N =

⊕P∈w-Max(R)

MP . Let i =∏iP be the canonical homomorphism

of M to∏

P∈w-Max(R)

MP . Then ker(i) = M , and hence i is injective if and only if

INJECTIVE MODULES OVER SM DOMAINS 359

M is co-semi-divisorial. Now we claim that i(M) ⊆ N . For let x ∈M . Then O(x)is a nonzero proper w-ideal of R, and is therefore contained in only finite numberof w-maximal ideals of R by [17, Theorem 4.9]. If O(x) * P ∈ w-Max(R), theelement x maps to 0 under the canonical mapping iP : M → MP (cf., see theremark after [2, Corollary 1.7]). Thus i(M) ⊆ N (Note that this also follows from[6, IX, Proposition 6.4]). Therefore we may assume that M is a submodule of N .Since N is semi-divisorial by [10, Corollary 9.7] and Corollary 3.6, it is sufficientto show that (N, i) is an essentially w-isomorphic extension of M by Proposition3.10. It is easy to see that i is an essential extension. Let Q be a w-maximal idealof R. Then by two well-known facts that the functor ⊗RRQ preserves direct sumsand (MP )Q = MP or 0 according as Q = P or not, we have NQ = MQ for anyw-maximal ideal Q of R. This implies that i is w-isomorphic by Lemma 3.11.

For the last assertion, first note that for any R-module M , M = 0 if and onlyif AssR(M) = ∅. Assume that there are two distinct P1, P2 ∈ AssR(M) such thatP1 = O(x1) and P2 = O(x2) for some x1, x2 ∈ M . By coirreducibility, there is0 6= y ∈M such that y ∈ Rx1∩Rx2. But then we have R = (< P1∪P2 >)w ⊆ (<O(x1)∪O(x2) >)w ⊆ O(y), and so 0 6= y ∈ M , which contradicts the assumptionthat M is co-semi-divisorial. �

We remark that if we specialize to Krull domains, then Theorem 3.12 yields[13, Theorem 4 (ii)] or [2, Proposition 1.9].

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Received March 7, 2006

Department of Information Security, Hoseo University, Asan 336-795, Korea

E-mail address: hkkim@hoseo.edu

Department of Mathematics, Kyungpook National University, Taegu 702-701, Ko-

rea

E-mail address: eskim@knu.ac.kr

Department of Mathematics, Kyungpook National University, Taegu 702-701, Ko-

rea

E-mail address: yngspark@knu.ac.kr