On primarily multiplication modules over pullback rings

The purpose of this paper is to present a new approach to the classification of indecomposable primarily multi-plication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [10] to a more general primarily multiplication modules ca...

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Дата:	2011
Автори:	Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani
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Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2011
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	On primarily multiplication modules over pullback rings / Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 2. — С. 1–17. — Бібліогр.: 30 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1547712019-06-16T01:31:28Z On primarily multiplication modules over pullback rings Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani The purpose of this paper is to present a new approach to the classification of indecomposable primarily multi-plication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [10] to a more general primarily multiplication modules case. 2011 Article On primarily multiplication modules over pullback rings / Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 2. — С. 1–17. — Бібліогр.: 30 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:13C05, 13C13, 16D70 http://dspace.nbuv.gov.ua/handle/123456789/154771 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The purpose of this paper is to present a new approach to the classification of indecomposable primarily multi-plication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [10] to a more general primarily multiplication modules case.
format	Article
author	Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani
spellingShingle	Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani On primarily multiplication modules over pullback rings Algebra and Discrete Mathematics
author_facet	Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani
author_sort	Reza Ebrahimi Atani
title	On primarily multiplication modules over pullback rings
title_short	On primarily multiplication modules over pullback rings
title_full	On primarily multiplication modules over pullback rings
title_fullStr	On primarily multiplication modules over pullback rings
title_full_unstemmed	On primarily multiplication modules over pullback rings
title_sort	on primarily multiplication modules over pullback rings
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/154771
citation_txt	On primarily multiplication modules over pullback rings / Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 2. — С. 1–17. — Бібліогр.: 30 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT rezaebrahimiatani onprimarilymultiplicationmodulesoverpullbackrings AT shahabaddinebrahimiatani onprimarilymultiplicationmodulesoverpullbackrings
first_indexed	2025-07-14T06:52:50Z
last_indexed	2025-07-14T06:52:50Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 11 (2011). Number 2. pp. 1 – 17 c© Journal “Algebra and Discrete Mathematics” On primarily multiplication modules over pullback rings Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani Communicated by D. Simson Abstract. The purpose of this paper is to present a new approach to the classification of indecomposable primarily multi- plication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [10] to a more general primarily multiplication modules case. 1. Introduction Modules arise when the representing subjects have an additive structure: representations as endomorphisms of abelian groups and vector spaces are the main cases. A representation of a group G over a field K is the same thing as a module over the corresponding group algebra K[G]. A module over a ring R is "really just" an abelian group M together with a ring morphism from to the endomorphism ring of M . It is an important feature of this strategy that any single representation will tell us only a certain amount about the original structure. So, for example, one looks at the set of all irreducible characters (simple modules) of a finite group, and even then for some purposes one has to look at more general modules. Thus arises the project of classifying all representations or, more realistically, all representations of a certain significant type. A commonly adopted strategy is to prove a decomposition theorem which says that every representation of the sort we are considering may be built up from certain simpler ones, 2000 Mathematics Subject Classification: 13C05, 13C13, 16D70. Key words and phrases: Pullback; Separated modules; Non-separated modules; Primarily multiplication modules; Dedekind domains; Pure-injective modules; Prüfer modules. 2 On primarily multiplication modules and then to develop a classification and structure theory for these simpler building blocks. An optimal structure theory for the blocks is one which provides us with a complete list and with representations of the members of the list, which are explicit enough to allow us to answer many questions about the blocks with relatively little effort. Of course it may be that, as in the case of finite groups, describing the ways in which the building blocks may be combined is problematic: fortunately with modules there are many structure theorems where the blocks are combined simply by forming direct sums. The structure theory for the building blocks is the theory of canonical forms for irreducible blocks. In fact many classification problems in linear algebra are most easily investigated by turning them into problems about modules. Also modules arise in various ways in analysis and topology, and classification may be relevant (see the discussion of Aronszajn and Fixman [2] in [26]). One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary rings R. The reader is referred to [1], [27, Chapters 1 and 14], [28] and [29] for a detailed discussion of classification problems, their representation types (finite, tame, or wild), and useful computational reduction procedures. Unfortunately, for the vast majority of rings, the classification of arbi- trary module is infeasible. For example, the classification of all indecompos- able pure-injective modules with infinite-dimensional top over R/rad(R) (for any module M over a ring R we define its top as M/rad(R)M) over the pullback ring formed by mapping two local Dedekind domains R1 and R2 onto a field R̄ is at least as difficult as that problem. Why consider pure-injective modules? Pure-injective modules are model-theoretically typical: for example classification of the complete theories of R-modules reduces to classifying the (complete theories of) pure-injectives. Also, for some rings the "small" (finite-dimensional, finitely generated ...) modules are classified and in many cases this classification can be extended to give a classification of the (indecomposable) pure-injective modules. Indeed, there is sometimes a strong connection between infinitely generated pure- injective modules and families of finitely generated modules [5]. One point of this paper is to introduce a subclass of pure-injective modules. Modules over pullback rings has been studied by several authors (see for example, [24],[14], [12], [11] and [13]). Notably, there is the monumental work of Levy [18], resulting in the classification of all finitely generated indecomposable modules over Dedekind-like rings. Common to all these classification is the reduction to a "‘matrix problem"’ over a division ring (see [28, Section 17.9] for background on matrix problems and their applications). In the present paper we introduce a new class of R-modules, R. E. Atani, Sh. E. Atani 3 called primarily multiplication modules (see Definition 3.1), and we study it in details from the classification problem point of view. We are mainly interested in case either R is a Dedekind domain or R is a pullback of two local Dedekind domains. Let R be the pullback of two local Dedekind domains over a common factor field. The purpose of this paper is to give a complete description of the indecomposable primarily multiplication mod- ules over R. The classification is divided into two stages: the description of all indecomposable separated primarily multiplication R-modules and then, using this list of separated primarily multiplication modules we show that non-separated indecomposable primarily multiplication R-modules are factor modules of finite direct sums of separated primarily multipli- cation R-modules. Then we use the classification of separated primarily multiplication modules from Section 2, together with results of Levy [18], [19] on the possibilities for amalgamating finitely generated separated modules, to classify the non-separated indecomposable primarily multipli- cation modules M (see Theorem 5.8). We will see that the non-separated modules may be represented by certain amalgamation chains of separated indecomposable primarily multiplication modules (where infinite length primarily multiplication modules can occur only at the ends) and where adjacency corresponds to amalgamation in the socles of these separated primarily multiplication modules. Several problems stated in the paper remain open, and we hope that this paper encourages the researchers to study the following two problems: (1) Describe the indecomposable primarily multiplication modules with finite-dimensional top over the pullback ring formed by mapping two local rings R1 and R2 onto a field R̄. (2) Describe the indecomposable primarily multiplication modules with finite-dimensional top over the pullback ring formed by mapping two local Dedekind domains R1 and R2 onto a semi-simple artinian ring R̄. 2. Preliminaries In this paper all rings are commutative with identity and all modules unitary. In order to make this paper easier to follow, we recall in this section various notions from module theory which will be in the sequel. Let v1 : R1 → R̄ and v2 : R2 → R̄ be homomorphisms of two local Dedekind domains Ri onto a common field R̄. Denote the pullback R = {(r1, r2) ∈ R1 ⊕ R2 : v1(r1) = v2(r2)} by (R1 v1−→ R̄ v2←− R2), where R̄ = R1/J(R1) = R2/J(R2). Then R is a ring under coordinate-wise multiplication. Denote the kernel of vi, i = 1, 2, by Pi. Then Ker(R → R̄) = P = P1 × P2, R/P ∼= R̄ ∼= R1/P1 ∼= R2/P2, and P1P2 = P2P1 = 0 4 On primarily multiplication modules (so R is not a domain). Furthermore, for i 6= j, 0→ Pi → R→ Rj → 0 is an exact sequence of R-modules (see [17]). Definition 2.1. An R-module S is defined to be separated if there exist Ri-modules Si, i = 1, 2, such that S is a submodule of S1 ⊕ S2 (the latter is made into an R-module by setting (r1, r2)(s1, s2) = (r1s1, r2s2)) [17]. Equivalently, S is separated if it is a pullback of an R1-module and an R2-module and then, using the same notation for pullbacks of modules as for rings, S = (S/P2S → S/PS ← S/P1S) [17, Corollary 3.3] and S ⊆ (S/P2S)⊕ (S/P1S). Also S is separated if and only if P1S ∩P2S = 0 [17, Lemma 2.9]. If R is a pullback ring, then every R-module is an epimorphic image of a separated R-module, indeed every R-module has a "minimal" such representation: a separated representation of an R-module M is an epimor- phism ϕ : S →M of R-modules where S is separated and, if ϕ admits a factorization ϕ : S f → S′ →M with S′ separated, then f is one-to-one. The module K = Ker(ϕ) is then an R̄-module, since R̄ = R/P and PK = 0 [17, Proposition 2.3]. An exact sequence 0→ K → S →M → 0 of R-modules with S separated and K an R̄-module is a separated representation of M if and only if PiS ∩K = 0 for each i and K ⊆ PS [17, Proposition 2.3]. Every module M has a separated representation, which is unique up to isomorphism [17, Theorem 2.8]. Moreover, R-homomorphisms lift to separated representation, preserving epimorphisms and monomorphisms [17, Theorem 2.6]. Definition 2.2. (a) If R is a ring and N is a submodule of an R-module M , the ideal {r ∈ R : rM ⊆ N} is denoted by (N : M). Then (0 : M) is the annihilator of M . A proper submodule N of a module M over a ring R is said to be primary submodule (resp. prime submodule) if whenever rm ∈ N , for some r ∈ R, m ∈M , then m ∈ N or rn ∈ (N : M) for some n (resp. m ∈M or rM ⊆ N), so rad(N : M) = P (resp. (N : M) = P ′) is a prime ideal of R, and N is said to be P -primary submodule (resp. P -prime submodule). The set of all primary submodules (resp. prime submodules) in an R-module M is denoted pSpec(M) (resp. Spec(M)) [20, 22]. (b) An R-module M is defined to be a multiplication module if for each submodule N of M , N = IM , for some ideal I of R. In this case we can take I = (N : M) [4]. (c) An R-module M is defined to be a weak multiplication module if Spec(M) = ∅ or for every prime submodule N of M , N = IM , for some ideal I of R (note that we can take I = (N : M)) [3]. R. E. Atani, Sh. E. Atani 5 (d) A submodule N of an R-module M is called pure submodule if any finite system of equations over N which is solvable in M is also solvable in N . A submodule N of an R-module M is called relatively divisible (or an RD-submodule) in M if rN = N ∩ rM for all r ∈ R [30]. (e) A module M is pure-injective if it has the injective property relative to all pure exact sequences [15, 30, 25]. (f) A non-zero R-module M is said to be coprimary if for each r ∈ R, the homothety M r. −→M is either injective or nilpotent. So rad(0 : M) = J , the radical (0 : M), is a prime ideal of R, and M is said to be J- coprimary [23]. Remark 2.3. (i) An R-module is pure-injective if and only if it is alge- braically compact [30, 25]. (ii) Let R be a Dedekind domain, M an R-module and N a submodule of M . Then N is pure in M if and only if N is an RD-submodule of M . Moreover, N is pure in M if and only if IN = N ∩ IM for each ideal I of R [30, 25]. (iii) It is easy to see that an R-module M is coprimary if and only if whenever rm = 0 (for r ∈ R, m ∈M), then either m = 0 or rnM = 0 for some n. Moreover, it is clear that if N is a J-primary submodule of M , then M/N is a J-coprimary R-module. 3. Basic properties of primarily multiplication modules The aim of this section is to classify primarily multiplication modules over a local Dedekind domain. First we collect some basic properties concerning primarily multiplication modules. We begin the key definition of this paper. Definition 3.1. An R-module M is defined to be a primarily multiplica- tion module if pSpec(M) = ∅ or for every primary submodule N of M , N = IM , for some ideal I of R One can easily show that if M is a primarily multiplication module, then N = (N : M)M for every primary submodule N of M . We need the following lemma proved in [23, p. 160 Theorem 10, p. 101 Corollary and p. 99 Corollary 1]. Lemma 3.2. (i) Let P be a prime ideal of R, let S be a multiplicatively closed set such that P ∩ S = ∅ and let M be an R-module. Then there exists a one-to-one correspondence between the P -primary submodules of M and the PS-primary submodules of MS. 6 On primarily multiplication modules (ii) Let K ⊆ N be submodules of an R-module M . Then N is a primary submodule of M if and only if N/K is a primary submodule of M/K. (iii) Let N be a P -primary submodule of the R-module M and suppose that I is an ideal of R and K a submodule of M . If IK ⊆ N , Then either I ⊆ P or K ⊆ N . Lemma 3.3. Let M be an R-module, N a P -primary submodule of M and I an ideal of R with I ⊆ (0 : M). Then N is P/I-primary submodule of M as an R/I-module. Proof. By Remark 2.3, M/N is a P -coprimary R-module. Let (a+ I)m ∈ N for some m ∈ M and a + I ∈ R/I, so a(m + N) = 0; hence either m ∈ N or asM ⊆ N for some s, as needed. Lemma 3.4. Let I be an ideal of a ring R, M a primarily multiplication R-module and N a non-zero R-submodule of M . Then the following hold: (i) If I ⊆ (N : M), then the R/I-module M/N is primarily multipli- cation. In particular, the R-module M/N is weakly multiplication. (ii) Every direct summand of M is a primarily multiplication R-module. Proof. (i) Let L be a primary submodule of M/N . Then by Lemma 3.2, there exists a primary submodule K of M such that L = K/N , so K = (K :R M)M . An inspection show that L = (L :R/I M/N)(M/N). Finally, take I = 0. Moreover, (ii) follows from (i). Lemma 3.5. Let R and R′ be any rings, f : R→ R′ a surjective homo- morphism and M an R′-module. Then the following hold: (i) If N is a primary R-submodule of M , then N is a primary R′- submodule of M . (ii) If M is a primarily multiplication R′-module, then M is a primarily multiplication R-module. Proof. (i) is clear. To see that (ii), assume that N is a primary R- submodule of M . Then N is primary R′-submodule of M , so N = IM for some ideal I of R. Set I ′ = f−1(I). Then f(I) = I ′ and IM = f(I)M = N , as required. Proposition 3.6. Let M be a module over a ring R. Then the following hold: (i) Let S be a multiplicatively closed subset of the ring R. If N is a primary submodule of M , then (N :R M)S = (NS :RS MS). (ii) M is a primarily multiplication R-module if and only if the RP - module MP is a primarily multiplication for every prime (or maximal) ideal P of R. R. E. Atani, Sh. E. Atani 7 Proof. (i) Let S ∩ rad(N : M) = ∅. Since the inclusion (N :R M)S ⊆ (NS :RS MS) is clear, we will prove the reverse inclusion. Assume that r/s ∈ (NS :RS MS) and let m ∈ M . Then (r/s)(m/1) ∈ NS , so utrm = sux ∈ N for some t, u ∈ S and x ∈ N . Therefore, N primary gives rm ∈ N ; hence r/s ∈ (N :R M)S , and so we have equality. So suppose that S∩rad(N : M) 6= ∅. Then obviously, NS = MS so that (N :R M)S = (NS :RS MS) = RS . (ii) Let M be a primarily multiplication R-module and N a primary submodule of MP , where P is a prime ideal of R. According to Lemma 3.2, we must have N = KP for some primary submodule K of M . So K = IM , therefore, N = (IM)P = IPMP . Conversely, let N be a primary submodule of M . It suffices to show that (N/(N : M)M)P = 0 for every maximal ideal P of R. If rad(N : M) ⊆ P , then by Lemma 3.2 again, NP is a primary submodule, so NP = (NP :RP MP ) = (N :R M)P by (i); hence (N/(N : M)M)P = 0. If rad(N : M) * P , then clearly NP = MP and (N : M)P = RP , so we have (N/(N : M)M)P = 0, as required. Remark 3.7. (1) Let R be an integral domain which is not a field and Q(R) the field of fractions of R. Since pSpec(Q(R)) = {0} by [11, Remark 2.7 (2)], we must have Q(R) is primarily multiplication. (2) Let R be a local Dedekind domain with unique maximal ideal P = Rp. As pSpec(E(R/P )) = ∅ (where E = E(R/P ) is the injective hull of R/P ) by [11, Remark 2.7 (b)], E is primarily multiplication. (3) It is easy to see that every primarily multiplication module over a ring R always is a weak multiplication module. Theorem 3.8. Let R be a local Dedekind domain with a unique maximal ideal P = Rp. Then the following is a complete list, up to isomorphism, of the indecomposable primarily multiplications modules: (i) R (ii) R/Pn, n ≥ 1, the indecomposable torsion modules; (iv) RP∞ = E(R/P ), the P -Prüfer module; (v) Q(R), the field of fractions of R. Proof. By [5, Proposition 1.3]) these modules are indecomposable. Clearly, R and R/Pn (n ≥ 1) are multiplication, so they are primarily multi- plication. Moreover, Q(R) and E(R/P ) are primarily multiplication by Remark 3.7. Now we show that there are no more indecomposable primarily multiplication R-modules. Now let M be an indecomposable primarily multiplication and choose any non-zero element a ∈ M . Let h(a) = sup{n : a ∈ PnM} (so h(a) is a nonnegative integer or ∞). Also let (0 : a) = {r ∈ R : ra = 0}: thus (0 : a) is an ideal of the form Pm or 0. Because (0 : a) = Pm+1 implies 8 On primarily multiplication modules that pma 6= 0 and p.pma = 0, we can choose a so that (0 : a) = P or 0. Now we consider the various possibilities for h(a) and (0 : a). Case 1. pSpec(M) = ∅. Since Spec(M) ⊆ pSpec(M), it follows from [11, Theorem 2.8 (Case 1)] M ∼= E(R/P ). So we may assume that pSpec(M) 6= ∅. Case 2. h(a) = n, (0 : a) = P . Say a = pnb. Then we have Rb ∼= R/Pn+1 is primarily multiplication. We also have Rb is pure in M . To see this, let r ∈ R: then r = pku for some integer k ≥ 0 and unit u ∈ R. If we have sb ∈ rM , say sb = pkuc, then write s = plt for some t ∈ R. So a = spn−1vb for some unit v ∈ R, that is a = uvpn−l+kc. Therefore k ≤ l and so we obtain sb = tu−1pl−k.pkb ∈ rM . Thus Rb is pure in M . Hence, since Rb is a pure submodule of bonded order of M , we obtain Rb is a direct summand of M by [16, Theorem 5]; hence M = Rb ∼= R/Pn+1. Case 3. h(a) = n, (0 : a) = 0. Say a = pnb. Then rb = 0 implies ra = 0 and so r = 0. Thus, Rb ∼= R. Furthermore Rb is pure in M by an argument like that in case (2). As M is a torsion-free R-module by [16, Theorem 10], Rb is prime (so primary) submodule of M (see [20, Result 2]); hence R ∼= Rb = P 0M = RM = M (since M 6= 0), as needed. Case 4. First we show that if h(a) = ∞, then (0 : a) = 0. Indeed, suppose not. Then (0 : a) = P . Since h(a) = ∞, there is an element a1 of M such that a = a0 = pa1, with a 6= a1 since a 6= 0 and pa = 0. If h(a1) < ∞, then by case (2), M is a module of finite length, and this contradicts the height of a is ∞. So a1 = pa2 for some a2 ∈M . By this process, one can show that M ∼= E(R/P ) (see [9, Proposition 2.7]); hence pSpec(M) = ∅ by Remark 3.7 (2), which is a contradiction. So we may assume that h(a) = ∞, (0 : a) = 0. By the Case (3) we may assume that every non-zero element of M satisfies these conditions. So a uniquely divisible by every non-zero element of Q(R) and so we can define a map, which is easily checked to be an R-homomorphism, from Q(R) to M which takes q to the element qa which, we have just shown, is defined and unique. Thus we have a copy of the injective module Q(R) embedded in M which must, therefore, be isomorphic to Q(R) (see [9, Proposition 2.7]). Corollary 3.9. Let R 6= M be a primarily multiplication module over a local Dedekind domain with maximal ideal P = Rp. Then M is of the form M = N ⊕K, where N is a direct sum of copies of R/Pn (n ≥ 1) and K is a direct sum of copies of E(R/P ) and Q(R). In particular, every primarily multiplication R-module not isomorphic with R is pure-injective. Proof. Let Ni denote the indecomposable summand of M . So by Lemma 3.4, Ni is an indecomposable primarily multiplication module. Now the assertion follows from Theorem 3.8 and [5, Proposition 1.3]. R. E. Atani, Sh. E. Atani 9 4. The separated case Throughout this section we shall assume unless otherwise stated, that R = (R1 v1−→ R̄ v2←− R2) (1) is the pullback of two local Dedekind domain R1, R2 with maximal ideals P1, P2 generated respectively by p1, p2, P denotes P1 ⊕ P2 and R1/P1 ∼= R2/P2 ∼= R/P ∼= R̄ is a field. In particular, R is a commutative Noetherian local ring with unique maximal ideal P . The other prime ideals of R are easily seen to be P1 (that is P1 ⊕ 0) and P2 (that is 0⊕ P2). In this section we determine the indecomposable primarily multiplica- tion separated R-modules where R is the pullback of two local Dedekind domains. Remark 4.1. Let R be the pullback ring as in (1), and let T be an R- submodule of a separated module S = (S1 f1 −→ S̄ f2 ←− S2), with projection maps πi : S ։ Si. Set T1 = {t1 ∈ S1 : (t1, t2) ∈ T for some t2 ∈ S2} T2 = {t2 ∈ S2 : (t1, t2) ∈ T for some t1 ∈ S1} Then for each i, i = 1, 2, Ti is an Ri-submodule of Si and T ≤ T1 ⊕ T2. Moreover, we can define a mapping π′ 1 = π1\|T : T ։ T1 by sending (t1, t2) to t1; hence T1 ∼= T/(0⊕Ker(f2)∩T ) ∼= T/(T ∩P2S) ∼= (T +P2S)/P2S ⊆ S/P2S. So we may assume that T1 is a submodule of S1. Similarly, we may assume that T2 is a submodule of S2 (note that Ker(f1) = P1S1 and Ker(f2) = P2S2). Theorem 4.2. Let R be the pullback ring as in (1), and let S = (S/P2S = S1 f1 −→ S̄ = S/PS f2 ←− S2 = S/P1S) be a separated R-module. Then S is a primarily multiplication R-module if and only if each Si is a primarily multiplication Ri-module (i = 1, 2). Proof. By [11, Remark 3.9], we may assume that pSpec(S) 6= ∅. Let S be a primarily multiplication R-module. Since (0⊕ P2) ⊆ ((0⊕ P2)S :R S), Lemma 3.3 gives S1 ∼= S/(0 ⊕ P2)S is a primarily multiplication R/(0⊕ P2) ∼= R1-module. Similarly, S2 is a primarily multiplication R2- module. Conversely, assume that each Si is a primarily multiplication Ri-module. Let T = (T1 → T̄ ← T2) be a primary submodule of S. We split the proof into two cases for rad(T : S). 10 On primarily multiplication modules Case 1. rad(T : S) = P . By [11, Proposition 3.3 (i)], we must have T2 = Pm 2 S2 ⊆ P2S2 for some m since S2 is primarily multiplication. Similarly, T1 = P k 1 S1 ⊆ P1S1 for some k. Let n = min{m, k}. Therefore, T ⊆ T1 ⊕ T2 ⊆ Pn−1(P1S1 ⊕ P2S2) = PnS. For the other containment, assume that s = (pn1 , p n 2 )(s1, s2) = (pn1s1, p n 2s2) ∈ PnS. Then s ∈ T since pn1s1 ∈ T1, pn2s2 ∈ T2 and f1(p n 1s1) = f2(p n 2s2) = 0 (note that Ker(f1) = P1S1 and Ker(f2) = P2S2). Thus T = PnS. Case 2. rad(T : S) = P1 ⊕ 0. Suppose that T is a (P1 ⊕ 0)-primary submodule of S. Then T2 is a 0-primary R2-submodule of S2 by [11, Proposition 3.3 (ii)]; hence T2 = 0 since (T2 : S2) ⊆ rad(T2 : S2) = 0. Therefore, T = (P1⊕ 0)S; hence S is a primarily multiplication R-module. Similarly, if rad(T : S) = 0⊕ P2, we get S is a primarily multiplication R-module. Lemma 4.3. Let R be the pullback ring as in (1), and let R 6= S = (S/P2S = S1 f1 −→ S̄ = S/PS f2 ←− S2 = S/P1S) be a separated R-module. If either S1 or S2 is torsion-free, then S̄ = 0. Proof. Assume that S̄ 6= 0 and let S1 be a torsion-free R1-module. Then P1S1 6= S1; hence P1S1 is a prime submodule of S1. By [20, Result 2], Pn+1 1 S1 = Pn 1 (P1S1) = P1S1 ∩ Pn 1 S1 = Pn 1 S1. Let s1 ∈ S1. Then there exists t1 ∈ S1 such that pn1 (s1 − p1t1) = 0; thus s1 = p1t1 ∈ P1S1. It follows that S1 = P1S1, which is a contradiction. Thus S̄ = 0. Lemma 4.4. Let R be the pullback ring as in (1). The following separated R-modules are indecomposable and primarily multiplication: (1) R = (R1 → R̄← R2) (2) S = (E(R1/P1) → 0 ← 0), (0 → 0 ← E(R2/P2) where E(Ri/Pi is the Ri-injective hull of Ri/Pi for i = 1, 2; (3) S = (Q(R1)→ 0← 0), (0→ 0← Q(R2) where Q(Ri) is the field of fractions of Ri for i = 1, 2; and, for all positive integers n,m, (4) S = (R1/P n 1 → R̄← R2/P m 2 ). Proof. By [5, Lemma 2.8] and [19, 1.9], these modules are indecomposable. Primarily multiplicativity follows from Theorem 3.8 and Theorem 4.2. We refer to modules of type (2) in Lemma 4.4 as P1-Prüfer and P2-Prüfer, respectively. R. E. Atani, Sh. E. Atani 11 Theorem 4.5. Let R be the pullback ring as in (1), and let S = (S1 f1 −→ S̄ f2 ←− S2 be an indecomposable separated primarily multiplication R- module. Then S is isomorphic to one of the modules listed in Lemma 4.4. Proof. We may assume that S 6= R. If pSpec(S) = ∅, then pSpec(Si) = ∅ by [11, Remark 3.9], so Si = PiSi for each i = 1, 2 by Theorem 3.8; hence S = PS = P1S1 ⊕ P2S2 = S1 ⊕ S2. Therefore, S = S1 or S2 and so S is of type (2) in the list of Lemma 4.4 by Theorem 3.8. So we may assume that pSpec(S) 6= ∅. Next suppose that PS = S. Then by [5, Lemma 2.7 (i)], S = S1 or S2 and so S is an indecomposable primarily multiplication Ri-modules for some i and, since PS = S, is type (3) in the list of Lemma 4.4 by Theorem 3.8. So we may assume that S/PS 6= 0. By Theorem 4.2, Si is a primarily multiplication Ri-module, for each i = 1, 2 (note that for each i, Si is torsion and it not divisible Ri-module by Theorem 1. Hence, by the structure of primarily multiplication modules over a local Dedekind domain (see Theorem 3.9), Si = Mi ⊕Ni where Ni is a direct sum of copies of Ri/P n i (n ≥ 1) and Mi is a direct sum of copies of E(Ri/Pi) and Q(Ri). Then we have S = (N1 → S̄ ← N2) ⊕ (M1 → 0 ← 0) ⊕ (0 → 0 ← M2). Since S is indecomposable and S/PS 6= 0 it follows that S = (N1 → S̄ ← N2). We will see that each Si (= Ni) is indecomposable. There exist positive integers m,n and k such that Pm 1 S1 = 0, P k 2 S2 = 0 and PnS = 0. Now choose t ∈ S1 ∪ S2 with t̄ 6= 0 and such that o(t) is maximal. There is a t = (t1, t2) such that o(t) = n, o(t1) = m and o(t2) = k. Then Riti is pure in Si for i = 1, 2 (see [5, Theorem 2.9]). Therefore, R1t1 ∼= R1/P m 1 (resp. R2t2 ∼= R2/P k 2 ) is a direct summand of S1 (resp. S2) since for each i, Riti is pure-injective [5]. Let M̄ be the R̄-subspace of S̄ generated by t̄. Then M̄ ∼= R̄. Let M = (R1t1 = M1 → M̄ ←M2 = R2t2). Then M is an R-submodule of S which is primarily multiplication by Lemma 4.4, and is a direct summand of S; this implies that S = M , and S is an in (4) (see [5, Theorem 2.9]). Corollary 4.6. Let R be the pullback ring as in (1), and let S 6= R be a separated primarily multiplication R-module. Then S is of the form S = M ⊕N , where M is a direct sum of copies of the modules as in (3), N is a direct sum of copies of the modules as in (1)-(2) of the Lemma 4.4. In particular, every separated primarily multiplication R-module not isomorphic with R is pure-injective. Proof. Apply Theorem 4.5 and [5, Theorem 2.9]. 12 On primarily multiplication modules 5. The non-separated case We continue to use the notation already established, so R is a pullback ring as in (1). In this section we find the indecomposable non-separated primarily multiplication modules with finite-dimensional top. It turns out that each can be obtained by amalgamating finitely many separated indecomposable primarily multiplication modules. Remark 5.1. Assume that R is the pullback ring as in (1) and let F = E(R/P ), the injective hull of R/P . Then pSpec(F ) = ∅ by [11, Proposition 4.1]; hence E(R/P ) is a non-separated primarily multiplication R-module (also see [5, p. 4053]). Proposition 5.2. Let R be the pullback ring as in (1) and let M be any primarily multiplication R-module. Then the following hold: (i) If M has a P1 ⊕ 0-primary submodule N , then M is separated. (ii) If M has a 0⊕ P2-primary submodule N , then M is separated. Proof. (i) First, we show that the (P1 ⊕ 0)-coprimary R-module M/N is separated. It suffices to show that (P1⊕0)(M/N) = 0. As (0, p2)(p1, 0)(m+ N) = 0 (m ∈M), we must have (p1, 0)m = 0. Thus M/N is a separated R-module. By assumption, (0⊕P2)N = (0⊕P2)(P n 1 ⊕0)M = 0 for some n; hence P1N ∩P2N = 0. Therefore, N is separated. Now we show that M is separated. It suffices to show that p1M ∩ p2M = 0. Let x = p1a = p2b for some a, b ∈M . Then p1(a+N) = p2(b+N) ∈ (P1(M/N))∩(P2(M/N)) = 0, so p1a = p2b ∈ N . Then N primary gives b ∈ N ; hence x = 0, and the proof is complete. The proof of (ii) is similar. Theorem 5.3. Let R be the pullback ring as in (1) and let M be any non-separated R-module. Let 0 → K → S → M → 0 be a separated representation of M . Then S is primarily multiplication if and only if M is primarily multiplication. Proof. By [11, Proposition 4.4], we may assume that pSpec(S) 6= ∅. If S is primarily multiplication, then M ∼= S/K is primarily multiplication, by Lemma 3.2. Conversely, assume that M is a primarily multiplication R-module and let N be a primary submodule of S. First suppose that rad(N : S) = P . Then by [11, Lemma 4.3], K ⊆ N , and N/K is a primary submodule of S/K ∼= M by Lemma 3.2, so Pn(S/K) = N/K for some n since S/K is primarily multiplication. As K ⊆ PnS (see [11, Lemma 4.3]), we must have N = PnS. If rad(N : S) = P1 ⊕ 0 (resp. rad(N : S) = 0 ⊕ P2), then N/K is a (P1 ⊕ 0)-primary (resp. (0 ⊕ P2)- primary) submodule of M , which is a contradiction by Proposition 2. Thus S is primarily multiplication. R. E. Atani, Sh. E. Atani 13 Proposition 5.4. Let R be the pullback ring as in (1) and let M be an indecomposable primarily multiplication non-separated R-module with finite-dimensional over R̄. Let 0 → K → S → M → 0 be a separated representation of M . Then S is pure-injective. Proof. By [5, Proposition 2.6 (i)], S/PS ∼= M/PM , so S has finite- dimensional top. Now the assertion follows from Corollary 4.6 and Theo- rem 5.3. Let R be the pullback ring as in (1) and let M be an indecomposable primarily multiplication non-separated R-module with M/PM finite- dimensional over R̄. Consider the separated representation 0→ K → S → M → 0. First we claim that R (the module (1) on the list in Lemma 4.4) is not a direct summand of S. For otherwise we have S = S′⊕R and then, since Soc(R) = 0, K ⊆ S′. Therefore M ∼= S′/K ⊕ R, a contradiction since M is indecomposable and non-separated. Moreover, By Proposition 5.4, S is pure-injective. So in the proofs of [5, Lemma 3.1, Proposition 3.2 and Proposition 3.4] (here the pure-injectivity of M implies the pure- injectivity of S by [5, Proposition 2.6 (ii)]) we can replace the statement "M is an indecomposable pure-injective non-separated R-module" by "M is an indecomposable primarily multiplication non-separated R-module": because the main key in those results are the pure-injectivity of S, the indecomposability and the non-separability of M . So we have the following results: Corollary 5.5. Let R be the pullback ring as in (1) and let M be an inde- composable primarily multiplication non-separated R-module with M/PM finite-dimensional over R̄, and let 0→ K → S →M → 0 be a separated representation of M . Then the quotient fields Q(R1) and Q(R2) of R1 and R2 do not occur among the direct summand of S. Corollary 5.6. Let R be the pullback ring as in (1) and let M be an inde- composable primarily multiplication non-separated R-module with M/PM finite-dimensional over R̄, and let 0→ K → S →M → 0 be a separated representation of M . Then S is a direct sum of finitely many indecompos- able primarily multiplication modules. Corollary 5.7. Let R be the pullback ring as in (1) and let M be an inde- composable primarily multiplication non-separated R-module with M/PM finite-dimensional over R̄, and let 0→ K → S →M → 0 be a separated representation of M . Then at most two copies of modules of infinite length can occur among the indecomposable summands of S. 14 On primarily multiplication modules Before we state the main theorem of this section let us explain the idea of proof. Let M be an indecomposable primarily multiplication non- separated R-module with M/PM finite-dimensional over R̄, and let 0→ K → S →M → 0 be a separated representation of M . Then by Corollary 5.6, S is a direct sum of just finitely many indecomposable separated primarily multiplication modules and these are known by Theorem 4.3. In any separated representation 0→ K i −→ S ϕ −→M → 0 the kernel of the map ϕ to M is annihilated by P , hence is contained in the socle of the separated module S. Thus M is obtained by amalgamation in the socle of the various direct summands of S. This explains Corollary 5.5: the modules Q(R1) and Q(R2) have zero socle and so cannot occur in a separated (hence "minimal") representation. So the questions are: does this provide any further condition on the possible direct summands of S? How can these summands be amalgamated in order to form M? In [19], Levy shows that the indecomposable finitely generated R- modules are of two non-overlapping types which he calls deleted cycle and block cycle types. It is the modules of deleted cycle type which are most relevant to us. Such a module is obtained from a direct sum, S, of indecomposable separated modules by amalgamating the direct summands of S in pairs to form a chain but leaving the two ends unamalgamated [19], see also [18, Section 11]. Since every "block cycle" type R-module is a quotient of a primarily multiplication R-module, so it is primarily multiplication by Lemma 3.2. So by Corollary 5.7, the infinite length non-separated indecomposable primarily multiplication modules are obtained in just the same way as the deleted cycle type indecomposable are except that at least one of the two "end" modules must be a separated indecomposable primarily multiplication of infinite length (that is, P1-Prüfer and P2-Prüfer). Note that one cannot have, for instance, a P1-Prüfer module at each end (consider the alternation of primes P1, P2 along the amalgamation chain). So, apart from any finite length modules we have amalgamations involving two Prüfer modules as well as modules of finite length (the injective hull E(R/P ) is the simplest module of this type), a P1-prüfer module and a P2-Prüfer module. If the P1-Prüfer and the P2-Prüfer are direct summands of S then we will describe these modules as doubly infinite. Those where S has just one infinite length summand we will call singly infinite (see [5, Section 3]). It remains to show that the modules obtained by these amalgamation are, indeed, indecomposable primarily multiplication modules. Theorem 5.8. Let R = (R1 → R̄ ← R2) be the pullback of two dis- crete valuation domains R1, R2 with common factor field R̄. Then the R. E. Atani, Sh. E. Atani 15 indecomposable non-separated primarily multiplication modules with finite- dimensional top are the following: (i) the indecomposable modules of finite length (apart from R/P which is separated); (ii) the doubly infinite primarily multiplication modules as described above; (iii) the singly infinite primarily multiplication modules as described above, apart from the two Prüfer modules (2) in Lemma 4.4. Proof. We know already that every indecomposable primarily multiplica- tion non-separated module with finite-dimensional top has one of these forms so it remains to show that the modules obtained by these amal- gamation are, indeed, indecomposable primarily multiplication modules. Let M be an indecomposable non-separated primarily multiplication R- module with finite-dimensional top and let 0→ K i −→ S ϕ −→M → 0 be a separated representation of M . (i) Since M is a quotient of a primarily multiplication R-module, so it is primarily multiplication by Lemma 3.2. The indecomposability follows from [19, 1.9]. (ii) and (iii) (involving one or two Prüfer modules) Since a quotient of any primarily multiplication R-module is primarily multiplication, M is primarily multiplication and the indecomposability follows from [5, Theorem 3.5]. Corollary 5.9. Let R be the pullback ring as described in Theorem 5.8. Then every indecomposable non-separated primarily multiplication R-module with finite-dimensional top is pure-injective. Proof. Apply [5, Theorem 3.5] and Theorem 5.8. Remark 5.10. Given a field k, the infinite-dimensional k-algebra T = k[x, y : xy = 0](x,y) is the pullback (k[x](x) → k ← k[y](y)) of the local Dedekind domains k[x](x), k[y](y). This paper includes the classification of those indecomposable primarily multiplication modules over T which have finite-dimensional top. References [1] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Vol. 1. 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Algebra 202 (2005), 118-132. [30] R. B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math., 28 (1969) 699-719. Contact information R. E. Atani Computer Engineering Department, University of Guilan, P.O. Box 3756, Rasht, Iran Sh. E. Atani Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran Received by the editors: 03.10.2010 and in final form 15.03.2011.

On primarily multiplication modules over pullback rings

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