Relative multiplication and distributive modules

a

Relative multiplication and distributive modules

Jos´e Escoriza, Blas Torrecillas

Abstract. We study the construction of new multiplication modules relative to a torsion theoryτ. As a consequence, τ-finitely generated modules over a Dedekind domain are completely determined. We relate the relative multiplication modules to the distributive ones.

Keywords: torsion theory, semicentered torsion theory, multiplication module, distribu- tive module

Classification: 13A15, 13G13

1. Introduction

Multiplications rings constitute an important class of rings and they have been studied by many authors (cf. [7], [8], [10], [18], [19], and [20]). They are generali- zations of Dedekind domains. Two concepts of multiplication module have been given. The first one was due to Singh and Mehdi (cf. [11]) and the second one, the most spread, was introduced by Barnard (cf. [2]). Multiplication modules have been recently considered by many authors, either over a commutative ring ([5], [9], [14] and their references) or over a noncommutative ring (cf. [13], [18], [19]

and [20]). Multiplication modules relative to a torsion theory have been defined and studied in [6] as a natural generalization of the absolute case.

The aim of this paper is to study the operations of relative multiplication modules in the commutative case. It is a work which will serve to research into the noncommutative case, which will be exposed in a subsequent paper. Section 2 is devoted to preliminaries and notation. We also include some results on relative multiplication ring and ideals. In [6], it was observed that every Krull domain with the canonical torsion theory is aτ-multiplicaton ring. Now, some examples of τ-multiplication rings which are not Krull domains are given. In Section 3, firstly, some properties for any hereditary torsion theory are found and are applied to find out if a module is or not relative multiplication. Then, operations such as intersection, sum, direct sum, multiplication, etc, between multiplication modules relative to a torsion theory have been studied. Finally, these results are applied to find out what modules over a Dedekind domain areτ-multiplication. In Section 4, relative distributive modules are introduced. Distributive modules have been studied in [1], [2], [4] and [17]. Relative distributive rings have been researched by Nˇastˇasescu (cf. [12]). Some elemental properties of relative distributive modules are shown. It is found the relationship between relative distributive modules

and relative multiplication modules in the main theorem. In the case of perfect torsion theories the distributive property of a module is characterized in terms of distributive property for its module of quotients with respect to the torsion theory.

2. Preliminaries and general notation

Throughout this paper,τis a hereditary torsion theory on a commutative ring R andM ∈R-Mod. The Gabriel filter associated to τ is denoted by F and the setSpec(R)− F is denoted byK(τ). A torsion theoryτ is semicentered(cf. [3], [16]) if for eachI /∈ F there exists a prime idealP such that P /∈ F and I⊆P. The ringR has enough τ-criticals if for every ideal I /∈ F of R there exists an idealP such that I⊆P andP is maximal with this condition. The set of such ideals is denoted byMax_F(R).

We shall give some easy properties of closure operations that will be useful for future results. IfS is a multiplicatively closed subset ofR, thenS⁻¹τ is the induced torsion theory byτ on the ring of quotientsS⁻¹R, whose Gabriel filter is{S⁻¹I;I ≤R}. IfP ∈Spec(R), then τP is the induced torsion theory inRP

with Gabriel filterF_P ={I_P;I∈ F}.

Let M, N be two R-modules. We denote by (M : N) ={r ∈R;r.N ⊆M} and by (M_P :N_P) ={x∈R_P;x.N_P ⊆M_P} whereP is any prime ideal ofR.

Rτ and Mτ represent the ring and the module of quotients with respect toτ respectively (cf. [16]).

The following lemma recollects some useful technical results. They are well- known and the proof is omitted.

Lemma 2.1. LetS be a multiplicatively closed subset of R. Let τ be a semi- centered torsion theory inR-Mod. Let P ∈K(τ). Let M,N be twoR-modules.

LetL≤M. Then

1. S⁻¹Cl_τ^M(L) =Cl^S_S−⁻¹1^Mτ (S⁻¹L);

2. Cl_τ^M_P^P(L_P) =L_P;

3. if Nisτ-finitely generated, then(M :N)_P = (M_P :N_P)for allP ∈K(τ);

4. if M is τ-finitely generated, then, for every P ∈ K(τ), (ann(M))_P = ann(MP).

Recall that an R-module M is called τ-multiplication if for every τ-closed submoduleN ofM there exists an idealIof Rsuch thatN =Cl^M_τ (I.M).

The definition of stronglyτ-multiplication module is a generalization of Singh and Mehdi’s definition (cf. [11]) for multiplication modules.

Definition 2.2. AnR-moduleM is called stronglyτ-multiplication if for allτ- closed submodulesN ⊆L, there exists an idealI ofRsuch thatN=Cl_τ^M(I.L).

A ringRis calledτ-multiplication if givenA,B τ-closed ideals ofRsuch that B⊆A, then there exists an idealI ofRverifyingB=Cl^R_τ(I.A).

Obviously, every ring Ris τ-multiplication as anR-module and it is strongly τ-multiplication as anR-module if and only if it is aτ-multiplication ring.

Example 2.3. ConsiderM =Z_p^∞ as aZ-module. Letτ be such that (p)∈ F. Since M is τ-simple, it is strongly τ-multiplication, but it is not multiplication according to Singh and Mehdi’s definition (cf. [11]).

Let R be an integral domain and let K be its field of quotients. Let τ be a torsion theory inR-Mod. A is a fractional ideal of R if there existsd∈R such thatd.A⊆R and it is anR-module.

Definition 2.4. A fractional idealA of R is calledτ-invertible if there exists a fractional idealB such thatCl^K_τ (A.B) =R.

Proposition 2.5. Everyτ-invertible ideal isτ-multiplication.

Proof: LetA be aτ-invertible ideal and B ⊆ A another ideal of R such that Cl^A_τ(B) =B. Then, there exists a fractional ideal C such thatCl_τ^K(A.C) =R.

Therefore, we haveB=Cl^A_τ(B.R) =Cl^K_τ (B.Cl^K_τ (A.C))∩A=Cl_τ^K(B.C.A)∩A= Cl^A_τ((B.C).A). Moreover, sinceB.C ⊆A.C, B.C is an ideal ofR.

IfAis an integral ideal andCl^R_τ(A.B) =Rfor some fractional idealB, thenAis aτ-multiplication ideal, i.e., it isτ-multiplication as anR-module. In particular, every ideal belonging to the Gabriel filter isτ-multiplication.

It follows immediately that the product of twoτ-invertible ideals isτ-invertible and therefore, it isτ-multiplication.

In [6] it is proved that a Krull domain with the canonical torsion theory is a τ-multiplication ring. The following example is a ring which is not multiplication but isτ-multiplication and is not a Krull domain.

Example 2.6. LetR = Π_i∈NR_i where R_i =Z₄. According to [8, Example 3], R is not a multiplication ring. Set S = ⊕_i∈NRi. Obviously, S² =S and it is possible to consider the Gabriel filter F = {A ≤ R;S ⊆ A}. If B is an ideal ofR, then Cl^R_τ(B) = (B : S) clearly. Denote bye_i the element of R which has the i-th coordinate equal to 1 and the others are 0. Ifx∈B and B is τ-closed, then each component x_i of xhas to verifyx_i.e_i ∈B. Letx= (x_i)_i∈N verifying the preceding condition. Ifs∈S, the producty.s can be seen as a finite sum of elements ofB and therefore, it belongs toB. This means thatB= Π_i∈N(B∩R_i).

Thusτ-closed ideals are ideals of the formB= Πi∈NBi with Bi ≤Z₄. LetA, B be τ-closed ideals ofR such thatA⊆B. Since Z₄ is a multiplication ring (it is uniserial), for everyi ∈N, there exists an idealCi ofZ₄ such thatAi =Ci.Bi. Consequently,A=Cl^R_τ(C.B), whereC= Π_i∈NC_i.

Remark 2.7. Notice that the ring of quotients with respect to τ is Rτ =Hom_R(S, R) = Π_i∈NHom_R(R_i, R) = Π_i∈NR_i=R.

SinceRτ is not a multiplication ring butRis aτ-multiplication ring, it is proved that Proposition 4.14 in [6] is not necessarily true ifτ is not perfect.

Example 2.8. IfRis aτ-multiplication ring andFis the corresponding Gabriel filter, thenR⊕Ris multiplication with respect to the torsion theory whose Gabriel filter is{(I, J);I, J ∈ F}. Therefore, ifDis a Krull domain andτ is the canonical torsion theory, then D⊕D is a relative multiplication ring and obviously, it is not a Krull domain.

In the relative noetherian case, the following characterization is immediate from [6, Theorem 4.18].

Proposition 2.9. If R is τ-noetherian and τ is semicentered, then R is a τ- multiplication ring if and only if R_P is a multiplication ring for eachP ∈K(τ).

Some examples of relative multiplication rings appear in [6] and other examples are obtained in forecoming sections.

3. Operations withτ-multiplication modules

IfP ∈Spec(R), then the set{x∈M;c.m= 0 for somec∈R−P} is denoted byT_P(M). AnR-moduleM is calledP-torsionifM =T_P(M).

The starting point is the following result, which appears in [6].

Proposition 3.1. Let τ be a semicentered torsion theory on R. M is a τ- multiplication module if and only if for allP ∈K(τ), M is P-torsion or c.M ⊆ Cl^M_τ (R.m)for somem∈M andc∈R−P.

Proposition 3.2. If R isτ-noetherian(τ-artinian)andM is aτ-multiplication module, thenM isτ-noetherian(τ-artinian).

Proof: Show that M has A.C.C. onτ-closed submodules. In fact, we consider N₁≤N₂ ≤. . . withN_i ≤M τ-closed (i∈I). ThenN_i =Cl^M_τ ((N_i :M).M) by [6, Lemma 3.11]. ButCl_τ^M(N_i :M) = (Cl_τ^M(N_i) :M) from [6, Proposition 2.7]

and therefore (Ni:M) is a τ-closed ideal for everyi∈I. Moreover, (N₁ :M)≤ (N₂ : M) ≤. . . . By hypothesis, there exists i such that (N_i : M) = (N_i+1 : M) = (N_i+2:M) =. . . and hence

Ni=Cl^M_τ ((Ni:M).M) =Cl_τ^M((N_i+1:M).M) =N_i+1=. . . and thereforeM isτ-noetherian. For the artinian case the proof is analogous.

The converse result is false. In fact, consider the ringZwhich isτ-noetherian for anyτ. Let M =Z⊕Z. M is τ-noetherian but it is not τ-multiplication for any torsion theoryτ different from the trivial one (cf. [6, Lemma 3.13]).

Example 3.3. LetM =Z[x1, x2,· · ·] be theZ-module consisting of all polyno- mials in infinite indeterminatesx1, x2,· · · By [16, Corollary VI.6.15], every torsion theory onZis semicentered. If τ is different from the trivial one, thenM is not τ-noetherian, obviously. By applying Proposition 3.2,M is notτ-multiplication.

Definition 3.4. Letτ,σbe torsion theories onRwith Gabriel filtersFτ andFσ

respectively. Thenτ∧σis the torsion theory whose Gabriel filter isFτ∩ Fσ. Proposition3.5. Letτ andσ be two hereditary torsion theories onR. If M ∈ Mod-Risτ andσ-multiplication, thenM is aτ∧σ-multiplication module.

Proof: LetN beτ∧σ-closed. ThenN isτ-closed andσ-closed. By hypothesis and by [6, Lemma 3.11], N =Cl_τ^M((N : M).M) = Cl^M_σ ((N : M).M). So, for everyn∈N there existIn∈ Fτ and Jn∈ Fσ such thatIn.n⊆(N :M).M and Jn.n⊆(N :M).M. ThenIn∩Jn∈ Fτ∩ Fσ verifying (In∩Jn).n⊆(N :M).M.

HenceN =Cl^M_τ∧σ((N :M).M).

Compare the next result with [14, Lemma 7].

Proposition 3.6. If M is aτ-multiplicationR-module andM =P

i∈IMi, then N =Cl^M_τ (P

i∈I(N ∩M_i))for eachN τ-closed submodule of M. Proof: SinceM isτ-multiplication module we have

N =Cl_τ^M((N :M).M) =Cl^M_τ ((N :M).(X

i∈I

Mi))⊆Cl_τ^M((N :M)X

i∈I

Mi).

ThusN⊆Cl_τ^M(P

i∈I(Mi∩N)). ThereforeN =Cl^M_τ (P

i∈I(N∩Mi)).

Proposition 3.7. Let τ be a semicentered torsion theory on R and let M = P

i∈ICl_τ^M(R.m_i) for some elements m_i ∈ M (i ∈ I). M is a τ-multiplication module if and only if there exists an ideal Ji (i ∈ I) such that Cl^M_τ (R.mi) = Cl^M_τ (Ji.M)for eachi∈I.

Proof: The necessity is clear.

Conversely, suppose the existence of such ideals J_i and let P /∈ F. If there exists i∈I such thatJ_i 6⊆P, then, by hypothesis,J_i.M ⊆Cl_τ^M(R.m_i). Hence, there exists c ∈Ji−P ⊆R−P such that c.M ⊆Cl^M_τ (R.mi). If not, we have Ji ⊆P for alli∈I. So Cl^M_τ (R.mi) =Cl^M_τ (Ji.M)⊆Cl_τ^M(P.M) for allmi ∈M andM =Cl^M_τ (P.M) by the hypothesis. Therefore, there exists an idealJi of R such that Cl^M_τ (R.m_i) =Cl^M_τ (J_i.M) =Cl^M_τ (J_i.Cl^M_τ (P.M)) =Cl_τ^M(P.J_i.M) = Cl^M_τ (P.Cl^M_τ (J_i.M)) = Cl^M_τ (P.m_i). Thus m_i ∈ Cl_τ^M(P.m_i). So, there exists Hi∈ F such thatHi.mi⊆P.miand moreoverHi6⊆P. Thus there existsh−p∈ R−P such that (h−p).mi= 0 and hencemi∈T_P(M). ThenR.mi ⊆T_P(M).

Obviously, Cl^M_τ (TP(M)) = TP(M). Thus Cl^M_τ (R.mi) ⊆TP(M) and therefore M =T_P(M). By Proposition 3.1,M isτ-multiplication.

It is straightforward from Proposition 3.7 that every τ-cyclic module is a τ- multiplication module.

Example 3.8. Consider the Z-module M = Z_p^∞ = {_p¹i +Z;i ∈ Z^∗} ⊂ Q/Z wherepis a prime. If (p)∈ F, thenM =Cl_τ^M(Z.(¹_p+Z)) and by Proposition 3.7, M is aτ-multiplication module.

If (p) ∈ F, then/ M is not τ-noetherian. By Proposition 3.2, it is not τ- multiplication.

Proposition 3.9. Let τ be a semicentered torsion theory on R. If I is a τ- multiplication ideal ofR and M is a τ-multiplication R-module, then I.M is a τ-multiplicationR-module.

Proof: Consider P ∈ K(τ). It is clear that if I = T_P(I) or M = T_P(M), then TP(I.M) = I.M. If I 6=TP(I) and M 6= TP(M), then there exist c, d ∈ R−P such that c.I ⊆ Cl^R_τ(R.a) and d.M ⊆ Cl^M_τ (R.m) for some a ∈ R and m ∈ M. Therefore c.d.I.M ⊆ Cl^R_τ(R.a).Cl_τ^M(R.m) ⊆ Cl^M_τ (R.a.m). Hence by

Proposition 3.1,I.M is aτ-multiplication module.

The next result answers the question of when the sum ofτ-multiplication mo- dules isτ-multiplication. It is the analogous one to [14, Theorem 2].

Theorem 3.10. Letτ be a semicentered torsion theory on R. Let Mi (i ∈ I) be a family ofτ-multiplicationτ-closed submodules of anR-moduleM such that M = P

i∈IM_i. Let A = P

i∈I(M_i : M). Then the following conditions are equivalent:

1. M is aτ-multiplication module;

2. Mi=Cl^M_τ ((Mi :M).M)for alli∈I;

3. ann(m) +A∈ F for allm∈M;

4. for every P ∈ K(τ) either M = TP(M) or there exist z ∈ ∪i∈IMi and c∈R−P such thatc.M ⊆Cl_τ^M(R.z).

Proof: 1 ⇒ 2 is clear. Now suppose 2 holds. Suppose that m ∈ M and ann(m) +A /∈ F. Since τ is semicentered, there exists P ∈ K(τ) such that ann(m) +A ⊆P. So, (Mi : M)⊆P for all i ∈I. Hence (Mi :M).M ⊆P.M and we haveM_i=Cl^M_τ ((M_i :M).M)⊆Cl^M_τ (P.M). ThusM =Cl_τ^M(P.M). As m∈M, thenm =x1+x2+· · ·+xn with xi ∈Mi for i∈ {1,2, . . . , n}. Since Mi is a τ-multiplication module, we have Cl_τ^M(R.xi) = Cl^M_τ (Bi.Mi) for some ideal Bi of R. Then, by the same argument as in the proof of Proposition 3.7, Cl^M_τ (R.xi) =Cl_τ^M(P.xi). Therefore there existsKi∈ F such that Ki.xi ⊆P.xi

for eachi ∈ {1,2, . . . , n}. Hence there exists c_i ∈R−P such thatc_i.x_i = 0 for eachi∈ {1,2, . . . , n}. Therefore there existsc∈R−P such that c.m= 0. But then,c∈ann(m) which contradictsann(m)⊆P. Thus 3 is satisfied.

3 ⇒ 4. Let P ∈ K(τ) and suppose that T_P(M) 6= M. Then there exists m∈M such that ann(m)⊆P. By condition 3,A6⊆P. Therefore, there exists i∈Isuch that (Mi :M)6⊆P. Hence there existsc∈R−P such thatc.M ⊆Mi. Moreover, M_i 6= T_P(M_i) because if not, thenc.M ⊆T_P(M_i) and M would be P-torsion. By Proposition 3.1, there exist c^′ ∈ R−P and y ∈ Mi such that c^′.Mi⊆Cl_τ^M(R.y). Thereforec.c^′.M⊆c^′.Mi ⊆Cl_τ^M(R.y) andc.c^′ ∈R−P.

4⇒1 by Proposition 3.1.

Remark 3.11. The result is still true ifM =Cl^M_τ (P

i∈IMi).

Corollary 3.12. Let τ be a semicentered torsion theory on R. Let Mi (i ∈ I) be a family of τ-multiplication τ-closed submodules of an R-module M. If P

i∈I(M_i:M)∈ F, thenM is aτ-multiplication module.

Proof: We have M = Cl^M_τ ((P

i∈I(M_i : M)).M) ⊆ Cl_τ^M(P

i∈I(M_i : M).M)

⊆ Cl_τ^M(P

i∈IMi). Since ann(m) +P

i∈I(Mi : M) ∈ F, it suffices to apply

Theorem 3.10 and Remark 3.11.

In these conditions we denote byA=P

i∈I(M_i :M).

Corollary 3.13. Letτ be a semicentered torsion theory onR. LetMi (i∈I) be a family ofτ-closed τ-multiplication finitely generated submodules of M. If M =P

i∈IM_i, thenM is aτ-multiplication module if and only if ann(M_i) +A∈ F.

Proof: Suppose thatM isτ-multiplication andM_i=hx₁, . . . , xni(ndepending oni). From Theorem 3.21,ann(xj) +A∈ F (1≤j≤n). Hence

[ann(x₁)∩ · · · ∩ann(xn)] +A⊇Πⁿ_j=1(ann(xj) +A)∈ F.

Thereforeann(M_i) +A= [∩ⁿ_j=1ann(x_j)] +A∈ F for alli∈I.

Now, suppose that ann(Mi) +A ∈ F for all i ∈ I. Let m ∈ M. Since M = P

i∈IM_i, m = m₁ +· · · +mr with m_j ∈ M_j (1 ≤ j ≤ r). Since ann(m_j)⊇ann(M_j),ann(m_i) +A∈ F for 1≤j≤n. Moreover,ann(m) +A= [∩ⁿ_j=1ann(mj)] +A⊇Πⁿ_j=1(ann(mj) +A)∈ F. Thereforeann(m) +A∈ F for allm∈M. By Theorem 3.10,M is aτ-multiplication module.

Example 3.14. LetM =⊕^∞_n=1Cpⁿ where Cpⁿ is the cyclic group of order pⁿ andpa prime integer. M is aZ-module. Every C_pⁱ is cyclic and therefore it is τ-multiplication.

If (p)∈ F, then every/ C_pi isτ-closed. Moreover, (C_pi :M) = 0. ThusA= 0.

It holds ann(C_pⁱ) +A = (pⁱ) ∈ F/ for all i ≥ 0. By Corollary 3.13, M is not τ-multiplication.

Suppose that (p)∈ F. Letx=xi₁⊕ · · · ⊕xin∈M where eachxij ∈C_p^j. We have (pⁱⁿ).x= 0. HenceM isτ-torsion. ThereforeM is τ-multiplication in this case.

Corollary 3.15. Let τ be a semicentered torsion theory on R. Let M = P

i∈IM_i, M_i being a τ-closed τ-multiplication finitely generated submodule of M for alli∈I. M isτ-finitely generated if and only if there exists a finite subset J ⊆Isuch thatP

i∈J(M_i:M)∈ F.

Proof: Since M is τ-finitely generated, there exists a finitely generated sub- moduleF of M such thatM =Cl_τ^M(F). Therefore there exists a finite subset J of I such that M = Cl^M_τ (P

i∈JMi). By Theorem 3.10 and Remark 4.12, ann(m) +P

i∈J(Mi : M) ∈ F for allm ∈ M, in particular for all m ∈ F. As F is finitely generated, it holds that ann(F) +P

i∈J(Mi : M) ∈ F. However, ann(F)⊆(Mi:M) for alli∈J. Hence P

i∈J(Mi:M)∈ F.

Conversely, suppose that P

i∈J(Mi : M) ∈ F for some finite subset J of I.

By Theorem 3.10, M is τ-multiplication. Moreover, M = Cl^M_τ ((P

i∈J(M_i : M)).M) =Cl^M_τ (P

i∈J(M_i:M).M) =Cl_τ^M(P

i∈JM_i). ThereforeM isτ-finitely

generated.

Corollary 3.16. Letτbe a semicentered torsion theory onR. LetK, L₁, . . . , Ln

be τ-closed submodules of M. If K, K +Li (1 ≤ i≤ n), L₁∩ · · · ∩Ln are τ- multiplication modules, thenK+ (L₁∩ · · · ∩Ln)is aτ-multiplication module.

Proof: LetP ∈K(τ). Call L=L₁∩ · · · ∩Ln. Clearly,Lisτ-closed. Suppose that T_P(K+L) 6= K+L. Then T_P(K+Li) 6= K+Li for each 1 ≤ i ≤ n.

ConsiderA = (K : (K+L_i)) + (L_i : (K+L_i)). By applying Theorem 3.10 to K+Li, we obtainA6⊆P as there existsm∈K+Li such thatann(m) +A⊆P. However,A = (K : Li) + (Li : K). Since (K : Li)⊆ (K : L), we deduce that (K : L) + (L_i : K) 6⊆ P for 1 ≤ i ≤ n. Hence (K : L) + (L : K) = (K : L) + [(L₁:K)∩ · · · ∩(Ln:K)]6⊆P. Therefore there existsc^′∈R−P such that c^′ =a₁+a₂ with a₁ ∈(K : L) and a₂ ∈(L:K). Thus there existsc ∈R−P (a₁ ora₂) such that c∈(K:L) orc∈(L:K). Hence c.L⊆K or c.K ⊆Land thereforec.(K+L)⊆K or c.(K+L)⊆L. By [6, Corollary 4.24], K+L is a

τ-multiplication module.

Corollary 3.17. Let τ be a semicentered torsion theory onR. If K, L are τ- closed submodules of an R-module M such that (K : L) + (L : K) ∈ F, then K+Lis aτ-multiplication module.

Lemma 3.18. Let τ be a semicentered torsion theory on R. Let N₁ and N₂ be τ-closed submodules of an R-module M. If N1, N2 and N1 +N2 are τ- multiplication, thenN₁∩N₂ is aτ-multiplication module.

Proof: LetP ∈K(τ). IfT_P(N₁∩N₂)6=N₁∩N₂, then it is clear thatT_P(N₁)6=

N1, TP(N2)6=N2 and TP(N1+N2)6=N1+N2. By Theorem 3.10, there exist x ∈ N1, y ∈ N2, z ∈ N1 ∪N2, c1, c2, c ∈ N such that c1.N1 ⊆ Cl^Nτ¹(R.x), c₂.N₂⊆Cl^Nτ²(R.y) andc.(N₁+N₂)⊆Clτ^N1+N2(R.z).

Supposez∈N₁(similarly ifz∈N₂). ThenClτ^N1(R.z)⊆Cl^Nτ^1+N2(N₁) =N₁. Moreover,c.y∈N2becausey∈N2, andc.y∈N1becausec.y∈c.(N1+N2)⊆N1. Therefore we havec₂.c.(N₁∩N₂)⊆c.Clτ^N2(R.y)⊆Clτ^N2(R.c.y).

On the other hand, it is obvious thatc₂.c.(N₁ ∩N₂)⊆ N₁. So, there exists c2.c∈R−P withc2.c.(N1∩N2)⊆Clτ^N2(R.c.y)∩N1=Clτ^N1∩N2(R.c.y) and by Proposition 3.1,N₁∩N₂ is aτ-multiplication module.

Theorem 3.19. Letτbe a semicentered torsion theory onR. LetN₁, . . . , N_kbe τ-closed submodules of anR-moduleM such that Ni+Nj is aτ-multiplication module for alli, j, such that1≤i < j≤k. Then

1. N₁+· · ·+N_k is aτ-multiplication module;

2. N₁, . . . , N_kareτ-multiplication modules if and only if N₁∩ · · · ∩N_k is a τ-multiplication module.

Proof: To prove the first part, it suffices to follow the proof of [14, Theorem 8]

with slight modifications. Proposition 3.1 and Theorem 3.10 are needed.

For the second part, we use induction on k. Suppose N₁, . . . , N_k are τ- multiplication modules. Consider theτ-multiplication moduleX =N₂∩ · · · ∩N_k.

By Corollary 3.16, N₁ +X is a τ-multiplication module and by Lemma 3.18, N1∩X is a τ-multiplication module.

LetP ∈K(τ). IfT_P(N₁+N_i) =N₁+N_i, then T_P(N₁) =N₁. Suppose that N1∩· · ·∩N_kis aτ-multiplication module. LetP∈K(τ). SupposeTP(N1+Ni)6=

N₁+Ni for all i∈ {2,3, . . . n}. By Theorem 3.10, there existui ∈N₁∪Ni and c_i∈R−P such thatc_i.(N_i+N₁)⊆Clτ^N1+Ni(R.u_i). If for somei,u_i∈N₁, then ci.N1 ⊆ Cl^Nτ¹(R.ui) and by Proposition 3.1, N1 is τ-multiplication. Ifui ∈ Ni

for every 2 ≤ i ≤ k, then we have c₂. . . c_k.N₁ ⊆ N₁ ∩ · · · ∩N_k as ci.N₁ ⊆ c_i.(N₁+N_i)⊆N_i (2 ≤i≤k). By [6, Corollary 4.24], N₁ is a τ-multiplication

module.

Corollary 3.20. Letτ a semicentered torsion theory onR. Let Ki (1≤i≤n) be a family ofτ-closed submodules of anR-moduleM which areτ-multiplication modules and such that Ki +Kj is τ-multiplication for 1 ≤ i < j ≤ n. Then (K₁ ∩ · · · ∩Km) + (K_m+1∩ · · · ∩Kn) is a τ-multiplication module for every positive integerm < n.

Proof: ConsiderL=Km+1∩· · ·∩K_n. By Theorem 3.19,Lis aτ-multiplication module. By Corollary 3.16,K_i+Lis a τ-multiplication module (1≤i≤n) and by Corollary 3.16 again,L+ (K1∩ · · · ∩Kn) is aτ-multiplication module.

Denote ˆMi=⊕_j6=iMj. Compare the next result with [5, Theorem 2.2].

Theorem 3.21. Let τ be a semicentered torsion theory on R. Let M an R- module such thatM =⊕i∈IMi whereM_i^′sareτ-closed submodules of M. Then M is a τ-multiplication module if and only if the two following conditions are satisfied:

1. M_i is aτ-multiplication module for eachi∈I;

2. for eachi∈I there exists an idealAi of R, such that M_i=Cl^M_τ (A_i.M_i)andA_i.Mˆ_i= 0.

Proof: Suppose that M is a τ-multiplication module. Then Mi ∼= M/Mˆi

and therefore it is a quotient of a τ-multiplication module. Thus M_i is a τ- multiplication module.

On the other hand, sinceMiisτ-closed, there exists an idealAi ofRsuch that Mi =Cl^M_τ (Ai.M) =Clτ^Mi(Ai.M). SoAi.M ⊆Mi. ButAi.M= (⊕_j∈IAi.Mj) =

⊕_j∈I(Ai.Mj) ⊆ Mi. Therefore Ai.Mj = 0 for all j 6= i and hence Ai.Mˆj = 0.

Moreover,Ai.M=Ai.Mi andMi=Cl^M_τ (Ai.Mi).

Suppose thatP ∈K(τ). IfM_i=T_P(M_i) for all i∈I, then for eachm∈M_i there exists c ∈ R−P such that c.m= 0. Hence, for each x∈ M there exists c∈R−P such thatc.x= 0. ThusT_P(M) =M.

Suppose that there exists j ∈ I such that Mj 6= TP(Mj). Then by Propo- sition 3.1, there exist c ∈ R−P and m ∈ Mj such that c.Mj ⊆ Cl^M_τ (R.m).

By condition 2, there exists an ideal Aj ≤ R such that Cl_τ^M(Aj.Mj) = Mj

and Aj.Mˆj = 0. We havec.Aj.Mj ⊆c.Cl_τ^M(Aj.Mj) = Mj.c ⊆Cl_τ^M(R.m). If

A_j ⊆P, thenM_j =Cl_τ^M(A_j.M_j)⊆Cl^Mτ ^j(P.M_j) and henceM_j =Cl^Mτ ^j(P.M_j).

Therefore Mj = TP(Mj), a contradiction. Thus there existsd ∈(R−P)∩Aj

such thatc.d.M ⊆c.d.(⊕_j∈IMj)⊆c.d.Mj⊆Cl_τ^M(R.m) and by Proposition 3.1,

M is aτ-multiplication module.

Corollary 3.22. Letτ be a semicentered torsion theory onR. LetM_i (i∈I) be a family of finitely generatedτ-closed modules such thatM =⊕_i∈IM_i. Then, M is a τ-multiplication module if and only if Mi is a τ-multiplication module andann(M_i) +ann( ˆM_i)∈ F for eachi∈I.

Proof: Suppose thatM isτ-multiplication module. By Theorem 3.21, the first condition,Mi is a τ-multiplication module for eachi∈I, is true. Suppose that there existsi ∈ I such that ann(Mi) +ann( ˆMi) ∈ F/ . Sinceτ is semicentered, there existsP ∈K(τ) such thatann(Mi) +ann( ˆMi)⊆P. From Theorem 3.21, there exists A_i ≤ R verifying M_i = Cl^M_τ (A_i.M_i) and A_i.Mˆ_i = 0. Therefore Ai ⊆ ann( ˆMi). Thus Ai ⊆ P. Hence Mi = Cl^M_τ (P.Mi). Since Mi is finitely generated, there exists c ∈ R−P such that c.M_i = 0, a contradiction because c∈ann(Mi)∩(R−P) =∅.

Conversely, it suffices to apply Corollary 3.13.

Corollary 3.23. Letτ be semicentered. LetM =M₁⊕ · · · ⊕Mnwhere M_i is a τ-closed τ-multiplication finitely generated module for1 ≤i≤n. ThenM is τ-multiplication if and only if ann(Mi) +ann(Mj)∈ F for all1≤i6=j≤n.

Proof: Suppose that M is τ-multiplication. Clearly ann( ˆMi) ⊆ ann(Mj), if j 6= i. By Corollary 3.22, ann(M_i) +ann( ˆM_i) ∈ F for 1 ≤ i ≤ n. Thus ann(Mi) +ann(Mj) contains an element of the Gabriel filter and therefore it belongs to the filter.

Suppose that the second part of the equivalence is true. Since ann( ˆM_i) =

∩_j6=iann(Mj),

ann(M_i)+ann( ˆM_i) =ann(M_i)+∩_j6=iann(M_j)⊇Π_j6=i[ann(M_i)+ann(M_j)]∈ F.

We can apply this corollary to find all finitely generatedτ-multiplication mo- dules over Dedekind domains.

Corollary 3.24. Finitely generated τ-multiplication modules over a Dedekind domain are just modules of the formM =Cl^M_τ (N)whereN is isomorphic to an ideal of Rand τ-cyclic modules.

Proof: Let R be a Dedekind domain. A finitely generated R-module M is of the form

M ∼=I₁⊕ · · · ⊕Ir⊕R/α₁⊕ · · · ⊕R/αn,

whereα_i⊆α_i+1 (1≤i≤n),I_j (1≤j≤n) is an ideal ofR and everyR/α_i is a cyclicR-module.

Since R is a commutative noetherian ring, by [16, Corollary VI.6.15], every torsion theory is semicentered.

Ifτ is trivial, then every module is τ-torsion. Thus every module isτ-multi- plication. Suppose thatτ is not trivial. Consider three cases.

Case A: n= 0.

Ifr= 1, thenM is a projective ideal. By [15, Theorem 1], it is multiplication.

If r ≥ 2, then I_j is τ-closed (1 ≤ j ≤ r) and τ-multiplication. Moreover, ann(I₁) +ann(I₂) = 0∈ F. By Corollary 3.23,/ M is notτ-multiplication.

Case B: α₁∈ F.

Assumer= 0. ThenM isτ-torsion and hence it isτ-multiplication.

Suppose thatr≥1. For eachx∈M we havex∈Cl^M_τ (I1⊕· · ·⊕Ir⊕0⊕· · ·⊕0).

LetN =I₁⊕ · · · ⊕Ir. By [6, Theorem 3.7], M is τ-multiplication if and only if N isτ-multiplication.

If r = 1, then N is a projective ideal and therefore it is τ-multiplication.

Consequently,M =Cl^M_τ (N) whereN is isomorphic to an ideal ofR.

If r > 1, then ann(I₁) +ann(I₂) = 0 ∈ F/ . By Corollary 3.23, N is not a τ-multiplication module.

Case C: α₁∈ F./ Suppose thatn= 1.

Ifr= 0, thenM =R/α₁is a cyclic module and therefore it is a multiplication module.

Now, assume r ≥ 1. We have ann(I₁) +ann(R/α₁) = α₁ ∈ F/ . By Corol- lary 3.23,M is notτ-multiplication.

Supposen≥2.

If there existsα_k (2≤k≤n) such that α_k∈ F, then α_k.x⊆Cl^M_τ (I₁⊕ · · · ⊕ Ir⊕R/α₁⊕ · · · ⊕R/α_k−1⊕0⊕ · · ·0). Let N = I₁⊕ · · · ⊕Ir⊕R/α₁⊕ · · · ⊕ R/α_k−1⊕0⊕ · · ·0. By [6, Theorem 3.7],M is τ-multiplication if and only if N isτ-multiplication.

Ifr≥2, thenann(I₁) +ann(I₂) = 0∈ F/ . Ifr= 1, then ann(I1) +ann(R/α1) =α1∈ F/ .

Ifr= 0 and k−1 = 1, thenN is cyclic and therefore it is multiplication. In this caseM isτ-cyclic.

Ifr = 0 andk−1 ≥2, then ann(R/α₁) +ann(R/α₂) =α₂ ∈ F/ . By Corol- lary 3.23,M is notτ-multiplication.

If none ofαj’s belongs to the Gabriel filter, the situation is absolutely similar

to the preceding one.

Immediately it follows the next corollary.

Corollary 3.25. Finitely generated τ-multiplication modules over a P.I.D. are justτ-cyclic modules.

Proposition 3.26. LetR,S be rings such that R⊆S. If X,Y areτ-multipli- cationR-modules insideS, thenX.Y is a τ-multiplicationR-module.

Proof: LetN =Cl_τ^X.Y(N). SinceXandY areτ-multiplication,Cl^X_τ (N∩X) = Cl^X_τ (I.X) and Cl^Y_τ(N ∩Y) = Cl^Y_τ(J.Y) for some I, J ≤ R. By applying the properties of the closure operation which appear in [6], we have Cl^X.Y_τ (N) = Cl^X.Y_τ (N∩X.Y) =Cl_τ^X.Y((N∩X).(N∩Y)) =Cl^X.Y_τ (Cl^X_τ (N∩X).Cl_τ^Y(N∩Y)) = Cl^X.Y_τ (Cl^X_τ (I.X).Cl_τ^Y(J.Y)) =Cl_τ^X.=Y(I.J.X.Y).

4. τ-distributive modules

AnR-moduleM is calleddistributiveif it has distributive property of the sum with respect to the intersection or distributive property of the intersection with respect to the sum, for the lattice of submodules.

Definition 4.1. A module M is called τ-distributive if the lattice of τ-closed submodules, denoted byCτ(M), is a distributive lattice.

The case M = R has been considered in [12]. Obviously, every distributive module is aτ-distributive module for anyτ. It is also immediate that every ring is τ-distributive if and only if it is τ-distributive as an R-module. If F = {R}

andτ the corresponding torsion theory, thenτ-distributive modules are just the distributive modules. If τ is perfect, then the R-module M is τ-distributive if and only if theRτ-moduleMτ is distributive. This is due to the isomorphism of lattices which appears in [16, Proposition 3.7].

Recall that aτ-torsionfree moduleM is calledτ-uniserial if its onlyτ-closed submodules areM and a chain (finite or infinite) of the form

0 =M₀⊂M₁⊂ · · · ⊂Mn⊂. . . Example 4.2. Everyτ-uniserial module isτ-distributive.

Theorem 4.3. Letτbe a semicentered torsion theory overR-Mod. The following sentences are equivalent:

1. M is aτ-distributiveR-module;

2. if N, L, K are submodules of M, then Cl^M_τ ((N +L)∩(N +K)) = Cl_τ^M(N+ (L∩K));

3. ifN,L, K are submodules of M, thenCl^M_τ (N∩(L+N)) =Cl^M_τ ((L+ N)∩(L+K));

4. M_P is distributive as anR_P-module for allP ∈K(τ);

5. (Rm:Rn) + (Rn:Rm)∈ F for allm, n∈M;

6. Cl_τ^M(R(m+n)) = Cl^M_τ ((Rm∩R(m+n)) + (Rn∩R(m+n))) for all m, n∈M;

7. Cl_τ^M(Rm+Rn) =Cl^M_τ (R(m+n) + (Rm∩Rn))for allm, n∈M; 8. Cl_τ^R((K+L) : N) = Cl^R_τ((K : N) + (L : N))for all K, L, N ≤ M, N

beingτ-finitely generated;

9. Cl_τ^R(K: (L∩N)) =Cl^R_τ((K :L) + (K:N))for allK, L, N ≤M, L, N beingτ-finitely generated;

10. Hom_RP((N/(N∩L))P,(L/(N ∩L))P) = 0 for allL, N ≤M and for all P ∈K(τ).

Proof: 1⇔2⇔3⇔4 it is similar to [12, Theorem 7.3].

1 ⇒ 5. Suppose that 1 is true. Letm, n ∈ M. By hypothesis, Cl^R_τ((Rm : Rn) + (Rn :Rm)) = R. Let P ∈ K(τ). Then Cl^Rτ_P^P((R_P^m₁ : R_Pⁿ₁) + (R_Pⁿ₁ : R_P^m₁)) =R_P. By Lemma 2.1, (R_P^m₁ : R_Pⁿ₁) + (R_Pⁿ₁ :R_P^m₁) = R_P. Hence (R_P^m_s :R_Pⁿ_t) + (R_Pⁿ_t :R_P^m_s) =R_P for all ^m_s,ⁿ_t ∈M_P. By [17, Theorem 1.6], MP is a distributiveRP-module for allP ∈K(τ). By 4,M isτ-distributive.

5⇒1. Conversely, suppose thatM isτ-distributive and (Rn:Rm) + (Rm: Rn) ∈ F/ . Since τ is semicentered, there exists P ∈ K(τ) such that (Rn : Rm) + (Rm: Rn) ⊆P. Thus [(Rm: Rn) : (Rn : Rm)]P ⊆ PP ⊂RP. Thus M_P is distributive as an R_P-module by Lemma 4.3 and by [17, Theorem 1.6], [(Rm:Rn) + (Rn:Rm)]_P =R_P, a contradiction.

6⇒5. By using [9, Lemma 3.1], we have

Cl^M_τ (R(m+n)) =Cl^M_τ ((Rm:R(m+n))(m+n)+(Rn:R(m+n))(m+n))

=Cl^M_τ ((Rm:R(m+n))+(Rn:R(m+n))(m+n)).

Sinceann(m+n)⊆(Rm:Rn) + (Rn:Rm), (Rm:Rn) + (Rn:Rm)∈ F.

1⇒6 is trivial.

7⇒5. By applying [9, Lemma 3.1], we haveCl^M_τ (Rm) =Cl^M_τ (Rm∩(Rm+ Rn)) =Cl^M_τ (((Rm:Rn) + (Rn:Rm))m).

Sinceann(m)⊆(Rn:Rm), 2 follows.

1⇒7. We haveRm⊆Rn+R(m+n). SinceM isτ-distributive,Cl^M_τ (Rm) = Cl^M_τ ((Rn∩Rm) + (R(m+n)∩Rm)). Analogously,Cl^M_τ (Rn) =Cl_τ^M((Rm∩ Rn) + (R(m+n)∩Rn)). Easily, it can be checked that Cl_τ^M(Rm+Rn) = Cl^M_τ ((Rm∩Rn) +R(m+n)).

1 ⇒ 8. Let P ∈ K(τ). By 4, M_P is distributive as an R_P-module. By [1, Theorem 1.9], ((KP+LP) :NP) = ((KP :NP)+(LP :NP)). SinceNP is finitely generated, we have ((K+L) : N)_P = ((K :N) + (L: N))_P for all P ∈K(τ).

Sinceτ is semicentered, 2 follows.

8 ⇒ 1. We shall prove that M_P is distributive as an R_P-module for each P ∈ K(τ). Let K_P, L_P, N_P ≤ M_P, N_P being finitely generated. Since N_P = h^x₁¹, . . . ,^x₁^ri, there existsN^′=hx₁, . . . , xni ≤N such that (Cl_τ^M(N^′))_P =N_P^′ = N_P and obviously, Cl_τ^M(N^′) is τ-finitely generated. From the hypothesis, by using localization, we obtain ((K+L) : N^′)_P = ((K :N) + (L :N))_P. Hence (K_P +L_P) : N_P = (K_P : N_P) + (L_P : N_P). By [1, Theorem 1.9], M_P is distributive.

8⇔9. From [1, Theorem 1.9], it suffices to use localization.

1⇔10 is straightforward by applying [17, Proposition 1.1].

Corollary 4.4. Let τ be a semicentered torsion theory onR. Let M be a τ- distributiveR-module. LetL,N be submodules of M. Then

1. if N is finitely generated andN∩L= 0, thenHom_R(N, L)isτ-torsion;

2. if M/Lis finitely generated andN+L=M, thenHom_R(M/L, M/N)is τ-torsion.

Proof: We shall prove 1. LetP ∈K(τ). By Theorem 4.3.10,Hom_RP(NP, LP)

= 0. Since N is finitely generated, the canonical morphism (HomR(N, L))P → Hom_RP(N_P, L_P) is injective. Thus (Hom_R(N, L))_P = 0 for allP∈K(τ). Hence 1 follows.

Now, prove 2. LetP ∈K(τ). By Theorem 4.3.10, we have Hom_RP((N/(N∩L))_P,(L/(N∩L))_P)

= 0∼=Hom_RP(((N+L)/L)_P,((N+L)/N)_P).

By hypothesis this module isHom_RP((M/L)_P,(M/N)_P). SinceM/L is finitely generated, (Hom_R(M/L, M/N))_P = 0 for allP ∈K(τ). Therefore, theR-module

Hom_R(M/L, M/N) isτ-torsion.

Example 4.5. Letτ be a semicentered torsion theory. By Theorem 4.3.4, every Krull domainR is aτ-distributive ring, i.e., it isτ-distributive as anR-module.

Proposition 4.6. If τis semicentered, then every submodule and every quotient of aτ-distributive module is aτ-distributive module.

Proposition 4.7. Letτ be a semicentered torsion theory overR-Mod. If M = Cl^M_τ (N), thenM isτ-distributive if and only if N isτ-distributive.

Proof: Suppose thatM is τ-distributive. By Theorem 4.3.4,MP is a distribu- tive R_P-module for all P ∈ K(τ). Then M_P = (Cl^M_τ (N))_P = Clτ^M_P^P(N_P) by Lemma 2.1. Moreover,M_P =N_P. Thus N_P is distributive as an R_P-module.

By Theorem 4.3.4 again,N isτ-distributive. The converse can be proved in the

same way.

Definition 4.8. A module is called τ-Bezout if everyτ-finitely generated sub- module isτ-cyclic.

Proposition 4.9. τ-distributive modules over a P.I.D. are justτ-Bezout modu- les.

Proof: Straightforward from Corollary 3.25.

The following results give different ways to obtain new relative distributive modules from relative distributive modules .

Proposition 4.10. Letτ be a semicentered torsion theory on R. LetM,N be twoτ-distributiveR-modules. Then

1. M ⊗_RN is aτ-distributiveR-module;

2. ifM is finitely generated, thenHom_R(M, N)is aτ-distributiveR-module.

Proof: By Theorem 4.3.4, 1 is trivial.

SinceM,N areτ-distributive modules,M_P, N_P are distributiveR_P-modules for all P ∈ K(τ). By [1, Lemma 4.1], Hom_RP(MP, NP) is distributive as an R_P-module. Since M is a finitely generated R-module, the canonical mor- phism [Hom_R(M, N)]_P → Hom_RP(M_P, N_P) is injective. By Proposition 4.6, [Hom_R(M, N)]_P is distributive for allP∈K(τ). By Theorem 4.3.4,Hom_R(M, N)

is aτ-distributiveR-module.

Remark 4.11. IfM isτ-finitely generated andN isτ-torsion free the same result is obtained. It suffices to realize that with the above hypothesis ifM =Cl^M_τ (F), then two maps belonging to [Hom_R(M, N)]_P which are equal over F are equal overM.

For anR-moduleM, setτ−Supp(M) ={P∈K(τ);M_P 6= 0}.

Proposition 4.12. LetMi (i∈I)be a family ofτ-distributives modules. Then

⊕_i∈IM_i isτ-distributive if and only if τ−Supp(M_i)∩τ−Supp(M_j) =∅for all i, j∈I i6=j.

Proof: Suppose that⊕_i∈IMi is τ-distributive and for somei6=j, there exists P ∈ K(τ) such that (M_i)_P 6= 0 6= (M_j)_P. The R_P-module (⊕_i∈I(M_i))_P ∼=

⊕i∈I(Mi)P is distributive. By [1, Proposition 1.8], Supp(Mi)∩Supp(Mj) = ∅.

HoweverP ∈Supp(M_i)∩Supp(M_j) =∅, a contradiction.

Conversely, letP ∈K(τ). If there existsQ.RP ∈Supp((Mi)P)∩Supp((M_j)P), then, since (M_i)_PQ.RP ∼= M_Q, (M_i)_Q 6= 0 6= (M_j)_Q for i 6= j. If Q ∈ F, then P ∈ F asQ⊆P. Therefore,Q∈K(τ), a contradiction.

The following theorem establishes a relationship betweenτ-distributive modu- les andτ-multiplication modules. It is a generalization of [2, Proposition 7].

Theorem 4.13. Let τ be a semicentered torsion theory on R-Mod. Then M is τ-distributive if and only if every τ-finitely generated submodule of M is τ- multiplication.

Proof: Suppose that M is τ-distributive. Let N = Cl_τ^N(F) ≤ M, F being finitely generated. By Theorem 4.3.4,M_P is distributive for all P ∈K(τ). Since F_P is finitely generated for allP ∈K(τ) as anR_P-module, by [2, Proposition 7], theR_P-moduleF_P is multiplication for allP ∈K(τ). By [6, Theorem 4.18],F is aτ-multiplicationR-module. By [6, Theorem 3.7],N isτ-multiplication as an R-module.

Conversely, suppose that every τ-finitely generated submodule of M is τ- multiplication. Let P ∈ K(τ). We shall prove that M_P is distributive as an R_P-module. Let N_P =h^x₁¹, . . . ,^x₁^ri with xi ∈ N (1 ≤i ≤ r). LetL_P ≤N_P.

ConsiderK =hx₁, . . . , xri ≤N. Obviously, K_P =N_P. Since Cl^N_τ (K)≤M is τ-finitely generated, it isτ-multiplication. HenceCl^N_τ (L∩N) =Cl_τ^N(I.Cl_τ^N(K)) for some idealI ofR. By localization,L_P =I_P.N_P. ThusN_P is multiplication

as anR_P-module.

Corollary 4.14. Let τ be a semicentered torsion theory. Every τ-noetherian τ-distributive module is a stronglyτ-multiplication module.

Corollary 4.15. If τis semicentered, then everyτ-uniserialτ-noetherian module M is stronglyτ-multiplication.

Corollary 4.16. Let τ be a semicentered torsion theory. Every τ-noetherian τ-distributive ring is aτ-multiplication ring.

In particular, ifτ is the canonical torsion theory, then every Krull domain is a τ-multiplication ring, by Corollary 4.16.

The following example shows that the τ-distributive modules class is strictly wider than the distributive modules class.

Example 4.17. LetR=K[x, y],Kbeing a field. Letτ be the canonical torsion theory. Since R is a Krull domain, it is a τ-multiplication ring. Thus every submodule of K is τ-multiplication. By Theorem 4.13, R is a τ-distributive R- module.

R is a integral domain which is not a Dedekind domain. By [10, Proposition 9.13], there exists some ideal (which must be finitely generated asRis noetherian) which is not multiplication as an R-module. By [2, Proposition 7], R is not distributive as anR-module.

Acknowledgments. This work will form part of the first author’s doctoral the- sis. The authors wish to thank Professor Patrick F. Smith (University of Glasgow) for his suggestions and ideas.

This work was started when the first author was an Erasmus student (P.I.C.

No. 93-B-1033/11) in the University of Glasgow.

The second author has been partially supported by the grant PB91-706 from DGICYT. We should also thank Professor C. Nˇastˇasescu for his comments about some parts of the work.

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Departamento de Algebra y An´alisis Matem´atico, Universidad de Almer´ıa, 04120 Almer´ıa, Spain

E-mail: [email protected] [email protected]

(Received June 10, 1996)