ABSORBING MULTIPLICATION MODULES OVER PULLBACK RINGS

Following some ideas and a technique introduced in [Comm. Algebra 41 (2013), pp. 776-791] we give a complete classification, up to isomorphism, of all indecomposable 2-absorbing multiplication modules with finitedimensional top over pullback of two discrete valuation domains with the same


Introduction
In this paper all rings are commutative with identity and all modules unitary.Let v 1 : R 1 → R and v 2 : R 2 → R be homomorphisms of two discrete valuation domains R i onto a common field R. Denote the pullback R = {(r 1 , r 2 ) ∈ R 1 ⊕ R 2 : R is a ring under coordinate-wise multiplication.Denote the kernel of v i , i = 1, 2, by P i .Then Ker(R → R) = P = P 1 × P 2 , R/P ∼ = R ∼ = R 1 /P 1 ∼ = R 2 /P 2 , and P 1 P 2 = P 2 P 1 = 0 (so R is not a domain).Furthermore, for i = j, 0 → P i → R → R j → 0 is an exact sequence of R-modules (see [20]).Modules over pullback rings has been studied by several authors (see for example, [1,5,9,15,19,25,32]).
Notably, there is the important work of Levy [22], resulting in the classification of all finitely generated indecomposable modules over Dedekind-like rings.Klingler [19] extended this classification to lattices over certain non-commutative Dedekindlike rings, and Haefner and Klingler classified lattices over certain non-commutative pullback rings, which they called special quasi triads, see [16,17].Common to all these classification is the reduction to a "matrix problem" over a division ring, see [6] and [29,Section 17.9] for a background of matrix problems and their applications.
Here we should point out that the classification of all indecomposable modules over an arbitrary unitary ring (including finite-dimensional algebras over an algebraically closed field) is an impossible task.In particular, an infinite-dimensional version of tame representation type is in fact wild representation type.For a discussion of this kind of problems the reader is referred to the papers by Ringel [28] and Simson [30].
The concept of 2-absorbing ideal, which is a generalization of prime ideal, was introduced and studied by Badawi in [2].Various generalizations of prime ideals are also studied in [3] and [4].Recall that a proper ideal I of a ring R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I.Recently (see [26,33]), the concept of 2-absorbing ideal is extended to the context of 2-absorbing submodule which is a generalization of prime submodule.
Recall from [26] that a proper R-submodule N of a module M is said to be a For the sake of completeness, we state some definitions and notations used throughout.Let R be the pullback ring as mentioned in the beginning of introduction.An R-module S is defined to be separated if there exist R i -modules S i , i = 1, 2, such that S is a submodule of S 1 ⊕ S 2 (the latter is made into an R-module by setting (r 1 , r 2 )(s 1 , s 2 ) = (r 1 s 1 , r 2 s 2 )).Equivalently, S is separated if it is a pullback of an R 1 -module and an R 2 -module and then, using the same notation for pullbacks of modules as for rings, S = (S/P 2 S → S/P S ← S/P 1 S) [20, Corollary 3.3] and S ⊆ (S/P 2 S) ⊕ (S/P 1 S).Also S is separated if and only if 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 (e) A submodule N of an R-module M is called a pure submodule if any finite system of equations over N which is solvable in M is also solvable in (ii) Let N be an R-submodule of M .It is clear that N is an RD-submodule of M if and only if for all m ∈ M and r ∈ R, rm ∈ N implies that rm = rn for some n ∈ N .Furthermore, if M is torsion-free, then N is an RD-submodule if and only if for all m ∈ M and for all non-zero r ∈ R, rm ∈ N implies that m ∈ N .In this case, N is an RD-submodule if and only if N is a prime submodule.

Basic properties of 2-absorbing multiplication modules
In this section, we give a complete description of the 2-absorbing multiplication modules over a discrete valuation domain.Our starting point is the following definition.Proof.By [8, Lemma 2.6], every non-zero proper submodule L of E is of the form ).Now we conclude that abSpec(E) = ∅.Thus E is a 2-absorbing multiplication module.
Clearly, 0 is a 2-absorbing submodule of Q(R).To show that 0 is the only 2absorbing submodule of Q(R), we assume the contrary and let N be a non-zero 2-absorbing submodule of Q(R).Since N is a non-zero submodule, there exists a/b, This contradicts the fact that N is a 2-absorbing submodule.Thus seSpec(Q(R)) = {(0)} and hence Q(R) is 2-absorbing multiplication.Finally, in the cases of R and R/P n these follows because they are multiplication modules.
Theorem 2.6.Let R be a discrete valuation domain with a unique maximal ideal P = Rp.Then the class of indecomposable 2-absorbing multiplication modules over R, up to isomorphism, consists of the following: (ii) R/P n , n ≥ 1, the indecomposable torsion modules; (iii) E(R/P ), the injective hull of R/P ; (iv) Q(R), the field of fractions of R. (ii) Assume that abSpec(S) = ∅ and let π be the projection map of R onto R 1 .
Conversely, assume that each S i is a 2-absorbing multiplication R i -module and let T = (T 1 → T ← T 2 ) be a 2-absorbing submodule of S. We may assume that (T : S) = 0.If (T : S) = P n 1 ⊕ P m 2 for some positive integers m, n, then S i = 0 for i = 1, 2, (T . Since T = P T it follows that for each i = 1, 2, T i is torsion and it is not divisible R i -module.Then there are positive integers m, n and k such that P m 1 T 1 = 0, P k 2 T 2 = 0 and P n T = 0.For t ∈ T , let o(t) denote the least positive integer m such that P m t = 0. Now choose t ∈ T 1 ∪T 2 with t = 0 and such that o(t) is maximal (given that t = 0).There exists [7,Theorem 2.9]).Thus, R 1 t 1 ∼ = R 1 /(0 : t 1 ) ∼ = R 1 /P m 1 is a direct summand of T 1 since R 1 t 1 is pure-injective; hence Let M be the R-subspace of T generated by t.Then M ∼ = R.Let M = (R 1 t 1 → M ← R 2 t 2 ).Then T = M , and T satisfies the case (iv) (see [7,Theorem 2.9]).
Theorem 3.5.Let S = R be an indecomposable separated 2-absorbing multiplication module over the pullback ring as in (1).Then S is isomorphic to one of the modules listed in Lemma 3.3.

The nonseparated case
We continue to use the notation already established, so R is the pullback ring as in (1).In this section we find the indecomposable non-separated 2-absorbing multiplication modules with finite-dimensional top.It turns out that each can be obtained by amalgamating finitely many separated indecomposable 2-absorbing multiplication modules.

2 -
absorbing submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N , then am ∈ N or bm ∈ N or ab ∈ (N : R M ).In the present paper we introduce a new class of R-modules, called 2-absorbing multiplication modules, and we study it in details from the classification problem point of view.We are mainly interested in case either R is a discrete valuation domain or R is a pullback of two discrete valuation domains.First, we give a complete description of the 2-absorbing multiplication modules over a discrete valuation domain.Let R be a pullback of two discrete valuation domains over a common factor field.Next, the main purpose of this paper is to give a complete description of the indecomposable 2-absorbing multiplication 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 2-absorbing multiplication R-modules and then, using this list of separated 2-absorbing multiplication modules we show that non-separated indecomposable 2-absorbing multiplication R-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable 2-absorbing multiplication R-modules.Then we use the classification of separated indecomposable 2-absorbing multiplication modules from Section 3, together with results of Levy[21,22] on the possibilities for amalgamating finitely generated separated modules, to classify the non-separated indecomposable 2-absorbing multiplication modules M with finitedimensional top (see Theorem 4.5).We will see that the non-separated modules may be represented by certain amalgamation chains of separated indecomposable 2-absorbing multiplication modules (where infinite length 2-absorbing multiplication modules can occur only at the ends) and where adjacency corresponds to amalgamation in the socles of these separated 2-absorbing multiplication modules.

[ 20 ,
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 Rmodule, since R = R/P and P K = 0 [20, Proposition 2.3].An exact sequence 0 → K → S → M → 0 of R-modules with S separated and K an R-module is a separated representation of M if and only if P i S ∩ K = 0 for each i and K ⊆ P S Proposition 2.3].Every module M has a separated representation, which is unique up to isomorphism [20, Theorem 2.8].Moreover, R-homomorphisms lift to a separated representation, preserving epimorphisms and monomorphisms [20, Theorem 2.6].Definition 1.1.(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 .Aproper submodule N of a module M over a ring R is said to be a 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 a P -prime submodule.The set of all prime submodules in an R-module M is denoted by Spec(M )[23,24].(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 : R M )[14].(c)A proper submodule N of a module M is said to be semiprime if whenever r k m ∈ N for some m ∈ M , r ∈ R, and positive integer k, then rm ∈ N .The set of all semiprime submodules in an R-module M is denoted by seSpec(M ).An R-module M is defined to be a semiprime multiplication module if seSpec(M ) = ∅ or for every semiprime submodule N of M , N = IM , for some ideal I of R[12].(d)A proper submodule N of a module M is said to be a 2-absorbing submodule if whenever a, b ∈ R, m ∈ M and abm ∈ N , then am ∈ N or bm ∈ N or ab ∈ (N : R M )[26,33].The set of all 2-absorbing submodules in an R-module M is denoted by abSpec(M ).

Definition 2 . 1 .Proposition 2 . 3 .Proposition 2 . 5 .
Let R be a commutative ring.An R-module M is defined to be a 2-absorbing multiplication module if abSpec(M ) = ∅ or for every 2-absorbing submodule N of M , N = IM , for some ideal I of R.One can easily show that if M is a 2-absorbing multiplication module, then N = (N : R M )M for every 2-absorbing submodule N of M .We need the following lemma proved in[33, Lemma 2.4] and[26,  Lemmas 2.1, 2.2, and Theorem 2.3], respectively.Lemma 2.2.(i) Let K ⊆ N be submodules of an R-module M .Then N is a 2-absorbing submodule of M if and only if N/K is a 2-absorbing submodule of M/K.(ii) Let I be an ideal of R and N be a 2-absorbing submodule of M .If a ∈ R, m ∈ M and Iam ⊆ N , then am ∈ N or Im ⊆ N or Ia ⊆ (N : M ).(iii) Let I, J be ideals of R and N be a 2-absorbing submodule of M .If m ∈ M and IJm ⊆ N , then Im ∈ N or Jm ⊆ N or IJ ⊆ (N : M ).(iv)Let N be a proper submodule of M .Then N is a 2-absorbing submodule of M if and only if IJK ⊆ N for some ideals I, J of R and a submodule Kof M implies that IK ⊆ N or JK ⊆ N or IJ ⊆ (N : M ).Let M be a 2-absorbing multiplication module over a commutative ring R. Then the following hold:(i) If I is an ideal of R and N a non-zero R-submodule of M with I ⊆ (N : M ), then M/N is a 2-absorbing multiplication R/I-module.(ii) If N is a submodule of M , then M/N is a 2-absorbing multiplication Rmodule.(iii)Every direct summand of M is a 2-absorbing multiplication submodule.(iv) If I is an ideal of R with I ⊆ (0 : M ), then M is a 2-absorbing multiplication R-module if and only if M is 2-absorbing multiplication as an R/I-module.Proof.(i) Let K/N be a 2-absorbing submodule of M/N .Then by Lemma 2.1 (i), K is a 2-absorbing submodule of M , so K = (K : M )M , where I ⊆ (N : M ) ⊆ (K : M ) = J.An inspection will show that K/N = (J/I)(M/N ).(ii) Take I = 0 in (i).(iii) Follows from (ii).(iv) It is easy to see that N is a 2-absorbing R-submodule of M if and only if N is a 2-absorbing R/I-submodule of M .Now the assertion follows the fact that (N : R M ) = (N : R/I M ).Remark 2.4.(i) Let R and R be any commutative rings, g : R → R a surjective homomorphism and M an R -module.It is clear that if N is a 2-absorbing Rsubmodule of M , then N is a 2-absorbing R -submodule of M .Suppose that M is a 2-absorbing multiplication R -module and let N be a 2-absorbing R-submodule of M .Then N = JM for some ideal J of R .It follows that I = g −1 (J) is an ideal of R with g(I) = J.Then IM = g(I)M = JM = N .Thus M is a 2-absorbing multiplication R-module.(ii) Let M be a 2-absorbing multiplication module over an integral domain R (which is not a field), and let T (M ) be the torsion submodule of M with T (M ) = M .Then T (M ) is a prime (so 2-absorbing) submodule M such that (T (M ) : M ) = 0 (see [24, Lemma 3.8]); hence T (M ) = 0. Thus M is either torsion or torsion-free.(iii) Let R = M = Z be the ring of integers.If N = 4Z, then N is a 2absorbing submodule of M , but it is not semiprime.So a 2-absorbing does not need to be semiprime.If K = 30Z, then an inspection will show that K is a semiprime submodule of M that it is not 2-absorbing.Hence a semiprime does not need to be 2absorbing.So the class of semiprime multiplication and 2-absorbing multiplication modules are different concepts.Let R be a discrete valuation domain with unique maximal ideal P = Rp.Then R, E = E(R/P ), the injective hull of R/P , Q(R), the field of fractions of R, and R/P n (n ≥ 1) are 2-absorbing multiplication modules.