Weak comultiplication modules over a pullback of commutative local Dedekind domains

The goal point of recent attempts to classify indecomposable modules over non-artinian rings has been pullback rings. The purpose of this paper is to outline a new approach to the classification of indecomposable weak comultiplication modules with finite-dimensional top over certain kinds of pullbac...

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Date:	2009
Main Authors:	Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani
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Published:	Інститут прикладної математики і механіки НАН України 2009
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Cite this:	Weak comultiplication modules over a pullback of commutative local Dedekind domains / Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani// Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 1–13. — Бібліогр.: 29 назв. — англ.

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spelling	irk-123456789-1533782019-06-15T01:26:20Z Weak comultiplication modules over a pullback of commutative local Dedekind domains Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani The goal point of recent attempts to classify indecomposable modules over non-artinian rings has been pullback rings. The purpose of this paper is to outline a new approach to the classification of indecomposable weak comultiplication modules with finite-dimensional top over certain kinds of pullback rings. 2009 Article Weak comultiplication modules over a pullback of commutative local Dedekind domains / Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani// Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 1–13. — Бібліогр.: 29 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 13C05, 13C13, 16D70. http://dspace.nbuv.gov.ua/handle/123456789/153378 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 goal point of recent attempts to classify indecomposable modules over non-artinian rings has been pullback rings. The purpose of this paper is to outline a new approach to the classification of indecomposable weak comultiplication modules with finite-dimensional top over certain kinds of pullback rings.
format	Article
author	Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani
spellingShingle	Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani Weak comultiplication modules over a pullback of commutative local Dedekind domains Algebra and Discrete Mathematics
author_facet	Reza Ebrahimi Atani Shahabaddin Ebrahimi Atani
author_sort	Reza Ebrahimi Atani
title	Weak comultiplication modules over a pullback of commutative local Dedekind domains
title_short	Weak comultiplication modules over a pullback of commutative local Dedekind domains
title_full	Weak comultiplication modules over a pullback of commutative local Dedekind domains
title_fullStr	Weak comultiplication modules over a pullback of commutative local Dedekind domains
title_full_unstemmed	Weak comultiplication modules over a pullback of commutative local Dedekind domains
title_sort	weak comultiplication modules over a pullback of commutative local dedekind domains
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/153378
citation_txt	Weak comultiplication modules over a pullback of commutative local Dedekind domains / Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani// Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 1–13. — Бібліогр.: 29 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT rezaebrahimiatani weakcomultiplicationmodulesoverapullbackofcommutativelocaldedekinddomains AT shahabaddinebrahimiatani weakcomultiplicationmodulesoverapullbackofcommutativelocaldedekinddomains
first_indexed	2025-07-14T04:36:14Z
last_indexed	2025-07-14T04:36:14Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2009). pp. 1 – 13 c© Journal “Algebra and Discrete Mathematics” Weak comultiplication modules over a pullback of commutative local Dedekind domains Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani Communicated by D. Simson Abstract. The goal point of recent attempts to classify indecomposable modules over non-artinian rings has been pullback rings. The purpose of this paper is to outline a new approach to the classification of indecomposable weak comultiplication modules with finite-dimensional top over certain kinds of pullback rings. 1. Introduction Let R = (R1 v1−→ R̄ v2←− R2) be the pullback of two semiperfect rings R1, R2 for which the factor rings R1/J(R1) and R2/J(R2) are isomorphic, say to R̄ ( V. V. Kirichenko named this ring the diad of the semiperfect rings R1 and R2 with common factor ring R̄ [12]). In [12], Kirichenko has given a description of the representations of the diad of generalized unise- rial algebras over algebraically closed field. It turns out that this problem is equivalent to the classification of pairs of mutually annihilating nilpo- tent operators acting in a space with gradation, and, in particular, the problem solved in [22] in connection with the study of irreducible rep- resentations of the Lorentz group is precisely the problem of describing representations of the diad of generalized uniserial algebras. His method was to reduce the problem to a matrix problem and then solve the ma- trix problem. The possibility of this reduction is based on the fact that one nilpotent operator acting in a graded space can be reduced to Jor- dan normal form. The problem of constructing all the finite metabelian 2000 Mathematics Subject Classification: 13C05, 13C13, 16D70. Key words and phrases: Pullback, Separated modules and representations, Non-separated modules, Weak comultiplication modules, Dedekind domains, Pure- injective modules, Prüfer modules. Jo u rn al A lg eb ra D is cr et e M at h .2 Weak comultiplication modules groups (that is, groups with an abelian commutator subgroup) is funda- mentally settled the Schreier theory of group extensions. Szekeres in [26], a more extensive class of metabelian groups were characterized by nu- merical invariants. In 1948 Brauer with his student K. A. Fowler, began the investigation of CA-groups of even order i.e. groups of even order in which the centralizer of every non-identity element is abelian. Around the same time, Wall began similar research at the suggestion of Graham Higman (there was some antecedents in the work of Szekeres [26]). Modules over pullback rings has been studied by several authors (see for example, [3], [22], [16], [29]). In the present paper we introduce a new class of R-modules, called weak comultiplication modules, the dual notion of weak multiplication modules, (see Definition 2.1), and we study it in details from the classification problem point of view (see [1], [24, Chapter 1] and [25, Chapter 19]). We are mainly interested in case either R is a Dedekind domain or R is a pullback of two local Dedekind domains. First, we give a complete description of the weak comultiplication modules over a Dedekind domain. Let R be a pullback of two local Dedekind domains over a common factor field. Next, the main purpose of this paper is to give a complete description of the indecomposable weak comultiplication R-modules with finite-dimensional top over R/rad(R) (for any module M we define its top as M/rad(R)M). The classification is divided into two stages: the description of all indecomposable separated weak comul- tiplication R-modules and then, using this list of separated weak comulti- plication modules we show that non-separated indecomposable weak co- multiplication R-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable weak comultiplication R-modules. Then we use the classification of separated indecomposable weak comultiplication modules from Section 3, together with results of Levy [16], [17] on the possibilities for amalgamating finitely generated separated modules, to classify the non-separated indecomposable weak comultiplication modules M with finite-dimensional top (see Theorem 4.8). We will see that the non-separated modules may be represented by certain amalgamation chains of separated indecomposable weak comulti- plication modules (where infinite length weak comultiplication modules can occur only at the ends) and where adjacency corresponds to amalga- mation in the socles of these separated weak comultiplication modules. There are a connection between the weak multiplication modules, comul- tiplication modules and the weak comultiplication modules. In fact, ev- ery indecomposable comultiplication R-module M is weak comultiplica- tion and every indecomposable weak comultiplication R-module M with finite-dimensional top is weak multiplication R-module, so they are pure- injectivr when M 6= R (see [10, 11, 5]). Jo u rn al A lg eb ra D is cr et e M at h .R. E. Atani, Sh. E. Atani 3 For the sake of completeness, we state some definitions and notations used throughout. In this paper all rings are commutative with identity and all modules unitary. Let v1 : R1 → R̄ and v2 : R2 → R̄ be ho- momorphisms 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 (so R is not a domain). Furthermore, for i 6= j, 0→ Pi → R→ Rj → 0 is an exact sequence of R-modules (see [15]). Definition 1.1. 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)). 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) [15, Corollary 3.3] and S ⊆ (S/P2S)⊕(S/P1S). Also S is separated if and only if P1S∩P2S = 0 [15, Lemma 2.9]. If R is a pullback ring, then every R-module is an epimorphic im- age of a separated R-module, indeed every R-module has a "minimal" such representation: a separated representation of an R-module M is an epimorphism ϕ = (S f → 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 [15, 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 [15, Proposition 2.3]. Every module M has a separated representation, which is unique up to isomorphism [15, Theo- rem 2.8]. Moreover, R-homomorphisms lift to a separated representation, preserving epimorphisms and monomorphisms [15, Theorem 2.6]. 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 prime submodule if whenever rm ∈ N , for some r ∈ R, m ∈ M , then m ∈ N or r ∈ (N : M), so (N : M) = P is a prime ideal of R, and N is said to be P -prime submodule. The set of all prime submodules in an R-module M is denoted Spec(M). Definition 1.2. (a) An R-module M is defined to be a weak multipli- cation module if Spec(M) = ∅ or for every prime submodule N of M , Jo u rn al A lg eb ra D is cr et e M at h .4 Weak comultiplication modules N = IM , for some ideal I of R (note that we can take I = (N : M)). (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). (c) An R-module M is defined to be a comultiplication module if for each submodule N of M , N = (0 :M I, for some ideal I of R. In this case we can take I = Ann(N) [2]. (d) We say that an R-module M is prime if the zero submodule of M is a prime submodule of M (so if N is a prime R-submodule of M , then M/N is a prime R-module). (e) 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 . So if N is pure in M , then IN = N ∩ IM for each ideal I of R. (f) 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. An important property of modules N, M over a Dedekind domains is that N is pure in M if and only if N is an RD-submodule of M (see [28] and [23] for more details). (g) A module M is pure-injective if it has the injective property relative to all pure exact sequences [28, 23]. In particular, by [14] and [28], an R-module is pure-injective if and only if it is algebraically compact. 2. Weak comultiplication modules over a Dedekind do- main The aim of this section is to classify weak comultiplication modules over a Dedekind domain. We begin the key definition of this paper. Definition 2.1. Let R be a commutative ring. An R-module M is defined to be a weak comultiplication module if Spec(M) = ∅ or for every prime submodule N of M , N = (0 :M I), for some ideal I of R. One can easily show that if M is a weak comultiplication module, then N = (0 :M Ann(N)) for every prime submodule N of M . We note that every comultiplication module is weak comultiplication module. Proposition 2.2. If R is a domain (not a field) and M is a weak comulti- plication R-module with torsion submodule T (M) 6= M , then Spec(M) = {T (M)}. Proof. Since T (M) is a prime submodule of M , it suffices to show that if N is a non-zero prime submodule of M , then N = T (M). By assumption, N = (0 :M I) for some non-zero ideal I of R, so N ⊆ T (M). For the Jo u rn al A lg eb ra D is cr et e M at h .R. E. Atani, Sh. E. Atani 5 reverse inclusion, assume that x ∈ T (M). Then rx = 0 ∈ N for some 0 6= r ∈ R; hence x ∈ N since (N : M) ⊆ (T (M) : M) = 0, and so we have equality. Note, if we assume the additional condition that 0 is a prime sub- module of M , then since 0 ⊆ T (M), (0 : M) ⊆ (T (M) : M) = 0 which implies 0 ∈ Spec(M) and so T (M) = 0. Proposition 2.3. Let M be a weak comultiplication module over a com- mutative ring R. Then the following hold: (i) If N is a pure submodule of M , then M/N is a weak comultipli- cation R-module. (ii) Every direct summand of M is a weak comultiplication submod- ule. In particular, if L is a direct summand of M , then M/L is a weak comultiplication R-module. Proof. (i) Let L/N be a prime submodule of M/N . Then by [21, Lemma 4.1], L is a prime submodule of M , so L = (0 :M I) for some ideal I of R; we show that L/N = (0 :M/N I). Let m + N ∈ (0 :M/N I). Then Im ⊆ N ; thus Im = Im ∩ N ⊆ IM ∩ N = IN = IL ∩ N = 0 (since N pure in M implies N is pure in L). Therefore, m ∈ L, and so (0 :M/N I) ⊆ L/N . The proof of the other inclusion is clear, and so we have equality. (ii) follows from (i) (since direct summands are pure). Proposition 2.4. Let M be a module over a Dedekind domain R. Then M is a weak comultiplication if and only if the RP -module MP is a weak comultiplication for every prime ideal P of R. Proof. Assume that M is a weak comultiplication R-module and let G be a prime submodule of MP , where P is a prime ideal of R. According to [18, Proposition 1], there exists a prime submodule N of M such that G = NP , so N = (0 :M J) for some ideal J of R. Therefore, G = NP = (0 :M J)P = (0 :MP JP ) by [27, Exercise 9.13]. The proof of the other implication is like that in [11, Proposition 2.3]. Proposition 2.5. Let M be a module over a Dedekind domain R. Then M is an indecomposable weak comultiplication R-module if and only if MP is indecomposable weak comultiplication as an RP -module for every prime ideal P of R. Proof. The proof is straightforward by Proposition 2.4. Reduction to the local case. Let R be a Dedekind domain. Our aim here is to classify weak comultiplication R-modules. By Proposition 2.5, it suffices to consider the case where R is a local Dedekind domain (i.e. a discrete valuation domain) with a unique maximal ideal P = Rp. Jo u rn al A lg eb ra D is cr et e M at h .6 Weak comultiplication modules Remark 2.6. Assume that R is a local Dedekind domain with maximal ideal P = Rp and let M = R (as a R-module). For a prime submodule PM of M we have (0 :M Ann(PM)) = R. Therefore, M is not a weak comultiplication R-module, but it is a weak multiplication R-module. Theorem 2.7. Let R be a discrete valuation domain with a unique max- imal ideal P = Rp. Then the following is a complete list, without repeti- tions, of the indecomposable weak comultiplication modules: (1) R/Pn, n ≥ 1, the indecomposable torsion modules; (2) RP∞ = E(R/P ), the P -Prüfer module; (3) Q(R), the field of fractions of R. Proof. First we note that each of the preceding modules is indecompos- able (by [5, Proposition 1.3]) and weak comultiplication. Since the only prime submodule of Q(R) is 0, by [19, Theorem 1], and 0 = (0 :Q(R) R), we must have that Q(R) is weak comultiplication. Moreover, as E(R/P ) is a torsion divisible R-module, we get Spec(E(R/P )) = ∅ by [20, Lemma 1.3]; hence it is weak comultiplication. In the case of R/Pn this follows because module R/Pn is comultiplication. Now let M be an indecomposable weak comultiplication 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 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. If Spec(M) = ∅, then by an argument like that in [9, Proposition 3.3 Case 1], we get M ∼= E(R/P ). So we may assume that Spec(M) 6= ∅. Case 2. If h(a) = n, then (0 : a) = P . Suppose not. Then (0 : a) = 0. Say a = pnb. Then rb = 0 implies ra = 0 and so r = 0. Thus Rb ∼= R and we also have that Rb is pure in M (see [9, Theorem 2.12 Case 1]). As M is a torsion-free R-module by [13, Theorem 10], we must have Rb is a prime submodule of M (see [18, Result 2]), so R ∼= Rb = (0 :M 0) = M , which is a contradiction by Remark 2.6. So we may assume that h(a) = n, (0 : a) = P . Say a = pnb. Then we have Rb ∼= R/Pn+1. Furthermore 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 [13, Theorem 5]; thus M = Rb ∼= R/Pn+1. Case 3. h(a) = ∞, (0 : a) = P . By an argument like that in [11, Theorem 2.5 Case 2], we get M ∼= E(R/P ), as needed. Case 4. h(a) = ∞, (0 : a) = 0. By an argument like that in [9, Theorem 2.12 Case3], we obtain M ∼= Q(R). Jo u rn al A lg eb ra D is cr et e M at h .R. E. Atani, Sh. E. Atani 7 Theorem 2.8. Let M be a weak comultiplication module over a discrete valuation domain with maximal ideal P = Rp. Then M is of the form M = N ⊕ K ⊕ L, where N is a direct sum of copies of the modules as described in (1), K is a direct sum of copies of the module as described in (2) and L is a direct sum of copies of the module as described in (3) of Theorem 2.7. In particular, every weak comultiplication R-module is pure-injective (also it is weak multiplication). Proof. Apply Proposition 2.3 (ii), Theorem 2.7, [5, Proposition 1.3] and [9, Theorem 3.5]. 3. 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 domains 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 weak comultiplica- tion separated R-modules where R is the pullback of two local Dedekind domains (we do not need the a priori assumption of finite-dimensional top for this classification). Theorem 3.1. Let R be a pullback ring as described in (1), and let S = (S1 → S̄ ← S2) be a separated R-module. Then S is a weak co- multiplication R-module if and only if each Si is a weak comultiplication Ri-module, i = 1, 2. Proof. Note that by [10, Theorem 2.7], Spec(S) = ∅ if and only if Spec(Si) = ∅ for i = 1, 2. So we may assume that Spec(S) 6= ∅. Assume that S is a separated weak comultiplication R-module and let 0 6= L be a non-zero prime submodule of S1. Then there exists a separated sub- module T = (T/P2S = T1 g1 −→ T̄ = T/PS g2 ←− T2 = T/P1S) of S, where gi is the restriction of fi over Ti, i = 1, 2 such that L = T1. We split the proof into three cases for (T : S). Case 1. (T : S) = P . Since T1 is a prime submodule of S1, we must have that T/(0⊕ P2)S is a prime R-submodule of S/(0⊕ P2)S; hence T is a prime R-submodule of S. By assumption, T = (0 :S Pn 1 ⊕ Pm 2 ) for Jo u rn al A lg eb ra D is cr et e M at h .8 Weak comultiplication modules some integers m, n; we show that T1 = (0 :S1 Pn 1 ). Let s1 ∈ (0 :S1 Pn 1 ). Then Pn 1 s1 = 0, so (Pn 1 ⊕ Pm 2 )(s1, 0) = 0; hence (s1, 0) ∈ T . Therefore, (0 :S1 Pn 1 ) ⊆ T1. Now suppose that x ∈ T1. Then there is an element y ∈ T2 such that g1(x) = g2(y), so (x, y) ∈ T ; hence Pn 1 x = 0, and so we have equality. Similarly, S2 is a weak comultiplication R2-module. Clearly, in this case, for each i = 1, 2, Si 6= 0. Case 2. (T : S) = P1 ⊕ 0. Since S is a weak comultiplication R- module, we have T = (0 :S Pn 1 ⊕ 0) for some positive integer n. By [10, Proposition 2.6], S1 = 0 and T2 is a 0-prime submodule of S2. Clearly, T2 = (0 :S2 R), and the proof is complete. The third case of (T : S) = 0⊕ P2 is similar. Conversely, assume that S1, S2 are weak comultiplication Ri-modules and let T be a non-zero prime submodule of S. If (T : S) = P , then for each i, Si 6= 0 and there exist positive integers n, m such that T1 = (0 :S1 Pn 1 ), T2 = (0 :S2 Pm 2 ), and so T = (0 :S Pn 1 ⊕ Pm 2 ). If (T : S) = P1 ⊕ 0, then S1 = 0 T2 = 0, and so T = (0 :S R). The case of (T : S) = 0⊕P2 is similar (see [10, Proposition 2.6]). Therefore S is a weak comultiplication R-module in every case. Lemma 3.2. Let R be a pullback ring as described in (1). The following separated R-modules are indecomposable and weak comultiplication: (1) 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; (2) S = (Q(R1)→ 0← 0) where Q(R1) is the field of fractions of R1; (3) (0→ 0← Q(R2) where Q(R2) is the field of fractions of R2; and, for all positive integers n, m, (4) S = (R1/Pn 1 → R̄← R2/Pm 2 ). Proof. By [5, Lemma 2.8], these modules are indecomposable. Weak comultiplicativity follows from Theorem 2.7 and Theorem 3.1 We refer to modules of type (1) in Lemma 3.2 as P1-Prüfer and P2- Prüfer respectively. Proposition 3.3. Let R be a pullback ring as described in (1), and let S be a separated weak comultiplication R-module. Then S is of the form S = M ⊕ N ⊕K, where M is a direct sum of copies of the modules as described in (1), N is a direct sum of copies of the modules described in 2) - 3) and K is a direct sum of copies of the modules described in (4) of the Lemma 3.2. In particular, every separated weak comultiplication R-module is pure-injective. Jo u rn al A lg eb ra D is cr et e M at h .R. E. Atani, Sh. E. Atani 9 Proof. Let T denote an indecomposable summand of S. Then we can write T = (T1 → T̄ ← T2), and T is a weak comultiplication R-module by Proposition 2.3. We split the proof into three cases. Case 1. If Spec(T ) = ∅, then Spec(Ti) = ∅ by [10, Theorem 2.7], so Ti = PiTi for each i = 1, 2 by Theorem 2.7; hence T = PT = P1T1 ⊕ P2T2 = T1 ⊕ T2. Therefore, T = T1 or T2 and so T is of type (1) in the list of Lemma 3.2 by Theorem 2.7. Case 2. If T has a (P1 ⊕ 0)-prime R-submodule U , then U/P1T is a 0-prime R2-submodule of the weak comultiplication T2 and T1 = 0 (so T̄ = 0) by [10, Proposition 2.6] and Theorem 3.1; hence T is of type (2) in the list of Lemma 3.2. Similarly, if T has a (0⊕P2)-prime R-submodule, then T is of type (3) in the list of Lemma 3.2. Case 3. If T has a P -prime R-submodule N = (N1 → N̄ ← N2), then PT ⊆ N 6= T , so PT 6= T (that is, T̄ 6= 0). Then by [10, Proposition 2.6] and Theorem 3.1, we must have P1T1 = N1 6= T1 and P2T2 = N2 6= T2; hence for each i = 1, 2, Ti is torsion and it is not divisible Ri-module (see Theorem 2.7). By an argument like that in [11, Proposition 3.3], we get T is a type (4) in the list of Lemma 3.2 by Theorem 2.7. Theorem 3.4. Let R be a pullback ring as described in (1), and let S be an indecomposable separated weak comultiplication R-module. Then S is isomorphic to one of the modules listed in Lemma 3.2. Proof. Apply Proposition 3.3 and Lemma 3.2. 4. The non-separated case We continue to use the notation already established, so R is a pullback ring as in (1). Proposition 4.1. Let R be a pullback ring as described in (1). Then E(R/P ) is a non-separated weak comultiplication R-module. Proof. By [10, Theorem 3.2], Spec(E(R/P )) = ∅. Therefore, E(R/P ) is a weak comultiplication R-module. Proposition 4.2. Let R be a pullback ring as described in (1) and let M be any weak comultiplication R-module. If M has either a P1 ⊕ 0-prime submodule or a 0⊕ P2-prime submodule, then M is separated. Proof. The proof is like that in [10, Proposition 3.4]. Theorem 4.3. Let R be a pullback ring as described in (1) and let M be any non-separated R-module. Let 0 → K → S → M → 0 be a separated Jo u rn al A lg eb ra D is cr et e M at h .10 Weak comultiplication modules representation of M . Then S is weak comultiplication if and only if M is weak comultiplication. Proof. By [10, Proposition 3.6], we may assume that Spec(S) 6= ∅. Sup- pose that M is a weak comultiplication R-module and let T be a non-zero prime submodule of S. Then by [10, Lemma 3.5], K ⊆ T , and so T/K is a prime submodule of S/K. By an argument like that in [11, Theorem 4.4], we get S is weak comultiplication. Conversely, assume that S is a weak comultiplication R-module and let N be a non-separated prime submodule of M . Then ϕ−1(N) = U is a prime submodule of S (see [7, Lemma 3.1]), so U = (0 :S Pn 1 ⊕ Pm 2 ) for some integers m, n. By [7, Lemma 3.1], U/K ∼= N is a prime submodule of S/K ∼= M , so an inspection will show that N = U/K = (0 :S/K Pn 1 ⊕Pm 2 ), as required. Proposition 4.4. Let R be a pullback ring as described in (1) and let M be an indecomposable weak comultiplication non-separated R-module with finite-dimensional top 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 is finite-dimen- sional top. Now the assertion follows from Theorem 4.3 and Proposi- tion 3.3. Let R be a pullback ring as described in (1) and let M be an inde- composable weak comultiplication non-separated R-module with M/PM finite-dimensional over R̄. Consider the separated representation 0 → K → S → M → 0. By Proposition 4.4, 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 weak co- 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 4.5. Let R be a pullback ring as described in (1) and let M be an indecomposable weak comultiplication 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 4.6. Let R be the pullback ring as described in (1) and let M be an indecomposable weak comultiplication non-separated R-module with M/PM finite-dimensional over R̄, and let 0 → K → S → M → 0 be a Jo u rn al A lg eb ra D is cr et e M at h .R. E. Atani, Sh. E. Atani 11 separated representation of M . Then S is a direct sum of finitely many indecomposable weak multiplication modules. Corollary 4.7. Let R be the pullback ring as described in (1) and let M be an indecomposable weak comultiplication 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. Recall that every indecomposable R-module of finite length is weak comultiplication (see Lemma 3.2 and Proposition 4.3). So by Corollary 4.7, the infinite length non-separated indecomposable weak comultiplica- tion 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 weak comultiplication 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 (the reader is re- ferred to [5], [10] and [11] for more details). It remains to show that the modules obtained by these amalgamations are indeed, indecomposable weak comultiplication modules. Theorem 4.8. Let R = (R1 → R̄ ← R2) be the pullback of two dis- crete valuation domains R1, R2 with common factor field R̄. Then the indecomposable non-separated weak comultiplication 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 weak comultiplication modules as described above, (iii) the singly infinite weak comultiplication modules as described above, apart from the two Prüfer modules (1) in Lemma 3.2. Proof. Let M be an indecomposable non-separated weak comultiplication R-module with finite-dimensional top and let 0 → K i −→ S ϕ −→ M → 0 be a separated representation of M . (i) Clearly, M is a weak comultiplication R-module. The indecom- posability follows from [17, 1.9]. Jo u rn al A lg eb ra D is cr et e M at h .12 Weak comultiplication modules (ii) and (iii) (involving one or two Prüfer modules) M is weak co- multiplication (see Proposition 3.3 and Proposition 4.1). Finally, the indecomposability follows from [5, Theorem 3.5]. Corollary 4.9. Let R be the pullback ring as described in Theorem 4.9. Then every indecomposable weak comultiplication R-module with finite- dimensional top is pure-injective. Proof. Apply [5, Theorem 3.5] and Theorem 4.8. Acknowledgement. The authors wish to express their sincere thanks to Professor Daniel Simson for his valuable suggestions. The authors also would like to thank the referee for several useful suggestions on the first draft of the manuscript. 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Wiseman, Projective modules over pullback rings, Math. Proc. Cam- bridge Philos. Soc. 97 (1985), 399-406. Contact information R. E. Atani Department of Electrical Engineering, Uni- versity of Guilan, P.O. Box 3756, Rasht, Iran Sh. E. Atani Department of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran Received by the editors: 22.03.2009 and in final form 01.05.2009.

Weak comultiplication modules over a pullback of commutative local Dedekind domains

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