ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING

The purpose of this paper is to introduce the concept of 2-absorbing comultiplication modules, which form a subclass of the class of pure-injective modules over pullback rings. A full description of all indecomposable 2absorbing comultiplication modules with finite-dimensional top over the pullback of two discrete valuation domains with the same residue field is given. Mathematics Subject Classification (2010): 03C45, 03C05, 16D70


Introduction
Over most rings it is impossible to classify all modules: even algebras of tame representation type typically are "wild" when their infinitely generated representations are considered.In particular, one is interested in the classification of certain "significant" modules rather than in arbitrary modules.The pure-injective modules seem to form a class of modules that appear naturally and where there is hope of some kind of classification.Pure-injective modules play a central role in the model theory of modules: for example classification of the complete theories of R-modules reduce 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 (see [31], [32], [33], [34] and [40]).The reader is referred to [3], [35,Chapters 1 and 14] and [36] for a detailed discussion of classification problems, their representation types (finite, tame, or wild), and useful computational reduction procedures.

EBRAHIMI, DOLATI, KHORAMDEL AND SEDGHI
In this paper all rings are commutative with identity and all modules unitary.
We are going to study pullbacks of discrete valuation rings.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 , where R = R 1 /J(R 1 ) = R 2 /J(R 2 ).Then 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 [2,Section 6]).Let R be a pullback of two discrete valuation domains with common residue field K.As in [24] define the associated graded ring G(R) to be the additive group ⊕ i P i /P i+1 equipped with a ring structure by defining the multiplication as in [24].Similarly, the associated graded module of an R-module M is P i M/P i+1 M , equipped with a G(R)-module structure by defining the scalar multiplication as in [24].Arnold and Laubenbacher ([2, Section 6]) showed that G(R) is the k-algebra k[x, y : xy = 0] (x,y) .The R-modules of deleted and block cycle types correspond exactly to the G(R)-modules of string and band types (see [8]).Modules over pullback rings have been studied by several authors (see for instance, [2], [7], [10], [12], [14], [15], [16], [17], [19], [20], [21], [23], [24], [25], [26], [29], [38]).Notably, there is the important work of Levy [25], 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 [9] and [35,Section 17.9] for a background of matrix problems and their applications.It is proved that the pullback of two commutative local rings has Morita duality if and only if the two rings have Morita duality ( [18]).
The classification of subclass of pure-injective modules over the pullback of two discrete valuation rings over a common factor field are very important and a difficult problem.One point of this paper is that to introduce a subclass of pure-injective modules over such rings.Indeed, this article includes the classification of those indecomposable 2-absorbing comultiplication modules over k[x, y : where k is a field, which have finite-dimensional top.
In the present paper, we introduce a new class of R-modules, called 2-absorbing comultiplication 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 comultiplication modules over a discrete valuation domain.Let R be a pullback of two discrete valuation domains over a common factor field.The main purpose of this paper is to give a complete description of the in- The concept of 2-absorbing ideal, which is a generalization of that of prime ideal, was introduced and studied by Badawi in [4].Various generalizations of prime ideals are also studied in [5] and [6].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 [30], [39]), the concept of 2-absorbing ideal is extended to the context of 2-absorbing submodule which is a generalization of prime submodule.
Recall from [30] 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 , (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 [31], [37]).
(g) A module M is pure-injective if it has the injective property relative to all pure exact sequences (see [31], [37]).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.

2-Absorbing comultiplication modules
In this section, we give a complete description of the 2-absorbing comultiplication modules over a discrete valuation domain.We begin with the key definition of this paper.(i) M/N is a 2-absorbing comultiplication R-module.
(ii) Every direct summand of M is a 2-absorbing comultiplication submodule.
Proof.(i) Let K/N be a 2-absorbing submodule of M/N .Then by Lemma 2.2(i), K is a 2-absorbing submodule of M , so K = (0 : M I) for some ideal I of R. We show that K/N = (0 : M/N I).Let x + N ∈ K/N .Then Ix = 0 gives I(x + N ) = 0; so x + N ∈ (0 : M/N I).For the reverse inclusion, assume that y + N ∈ (0 : M/N I).
Then Iy ⊆ N ∩ IM = IN ⊆ IK = 0; hence y ∈ K, and we have equality.
Remark 2.5.Let R be a discrete valuation domain with unique maximal ideal (c) Each R/P n (n ≥ 1) is a 2-absorbing comultiplication module since it is a comultiplication module (see [14]).
Theorem 2.6.Let R be a discrete valuation domain with a unique maximal ideal P = Rp.Then the indecomposable 2-absorbing comultiplication modules over R, up to isomorphism, are the following: (i) R/P n , n ≥ 1, the indecomposable torsion modules; (ii) E(R/P ), the injective hull of R/P ;  Theorem 2.7.Let M be a 2-absorbing comultiplication module over a discrete valuation domain with a unique maximal ideal P = Rp.Then M is of the form where N is a direct sum of copies of R/P n (n ≥ 1) and K is a direct sum of copies of E(R/P ) and Q(R).In particular, every 2-absorbing comultiplication R-module is pure-injective.
Proof.Let T denote an indecomposable summand of M .Then T is an indecomposable 2-absorbing comultiplication module by Proposition 2.4(ii).Now the assertion follows from Theorem 2.6.The "in particular" statement follows from [10, Proposition 1.3].

The separated absorbing comultiplication modules
In this section we determine the indecomposable absorbing comultiplication separated R-modules where is the pullback of two discrete valuation domains R 1 , R 2 with maximal ideals P 1 , P 2 generated respectively by p 1 , p 2 , P denotes P 1 ⊕P 2 and R 1 /P 1 ∼ = R 2 /P 2 ∼ = R/P ∼ = R is a field (we do not need the a priori assumption of finite-dimensional top for this classification).Then R is a commutative Noetherian local ring with unique maximal ideal P .The other prime ideals of R are easily seen to be P 1 (that is P 1 ⊕ 0) and P 2 (that is 0 ⊕ P 2 ).Let a = (r, s) ∈ R with r = 0 and s = 0. Then we can write a = (p n 1 , p m 2 ) for some positive integers m, n, so ann(a) = 0; hence Ra ∼ = R.If a = (0, p m 2 ) for some positive integer m, then ann(a) = P 1 ⊕ 0, and so The other ideals I of R are of the form I = P n 1 ⊕ P m 2 = (P n 1 , P m 2 ) = (< p n 1 >, < p m 2 >) for some positive integers m, n since I ⊆ P = P 1 ⊕ P 2 = (P 1 , P 2 ) = (< p 1 >, < p 2 >) and p 1 p 2 = 0 = p 2 p 1 (see [10, p. 4054]).We need the following lemma proved in   Set (ii) S is a direct sum of finitely many indecomposable 2-absorbing comultiplication modules.
(iii) At most two copies of modules of infinite length can occur among the indecomposable summands of S.
Before embarking on the proof of the next result let us explain its idea.Let R be a pullback ring as in (1).Let M be any R-module and let 0 → K → S → M → 0 be a separated representation of M .We have already shown that if M is indecompos-

decomposable 2 -
absorbing 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 2-absorbing comultiplication R-modules and then, using this list of separated 2-absorbing comultiplication modules we show that non-separated indecomposable 2-absorbing comultiplication R-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable 2-absorbing comultiplication R-modules.Then we use the classification of separated indecomposable 2-absorbing comultiplication modules from Section 3, together with results of Levy[25,26] on the possibilities of amalgamating finitely generated separated modules, to classify the non-separated indecomposable 2-absorbing comultiplication modules with finitedimensional top (see Theorem 4.7).We will see that the non-separated modules may be represented by certain amalgamation chains of separated indecomposable 2-absorbing comultiplication modules (where infinite length 2-absorbing comultiplication modules can occur only at the ends) and where adjacency corresponds to amalgamation in the socles of these separated 2-absorbing comultiplication modules.

[ 24 ,
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)[24, Corollary   3.3]  and S ⊆ (S/P 2 S) ⊕ (S/P 1 S).Also S is separated if and only if P 1 S ∩ P 2 S = 0 Lemma 2.9].Let M be an R-module.A separated representation of M is a pair (S, ϕ) where (i) S is a separated R-module;(ii) ϕ is an R-homomorphism of S onto M ;(iii) for every pair (S , ϕ ) satisfying (i) and (ii), and for every R-homomorphismα of S in S such that ϕ α = ϕ, α is 1 − 1.The module K = Ker(ϕ) is then an R-module, since R = R/P and P K = 0 [24, 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 [24, Proposition 2.3].Every module M has a separated representation, which is unique up to isomorphism [24, Theorem 2.8].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 .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 by Spec(M )[27,28].(b)An R-module M is a comultiplication module provided for each submodule N of M there exists an ideal I of R such that N is the set of elements m in M such that Im = 0.In this case we can take N = (0 : M ann(N ))[1].(c)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[15].(d) A proper submodule N of a module M is said to be 2-absorbing submodule if whenever a, b ∈ R, m ∈ M and abm ∈ N , then am ∈ N or bm ∈ N or ab ∈ (N : R M )[30,39].The set of all 2-absorbing submodules in an R-module M is denoted by abSpec(M ).

Remark 1 . 2 .
(a) 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 IN = N ∩ IM for each ideal I of R.Moreover, N is pure in M if and only if N is an RD-submodule of M [31], [37].(b) 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

Definition 2 . 1 .Proposition 2 . 4 .
Let R be a commutative ring.An R-module M is defined to be a 2-absorbing comultiplication module if abSpec(M ) = ∅ or for every 2-absorbing submodule N of M , N = (0 : M I), for some ideal I of R.One can easily show that if M is a 2-absorbing comultiplication module, then N = (0 : M ann(N )) for every 2-absorbing submodule N of M .It is easy to see that the class of 2-absorbing comultiplication modules contains the class of weak comultiplication modules (resp.comultiplication modules) defined in[15] (resp.[14]).We need the following lemma proved in[39, Lemma 2.4] and[30,  Lemma 2.1, Lemma 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 K of M implies that IK ⊆ N or JK ⊆ N or IJ ⊆ (N : M ).Lemma 2.3.(a) Let M be an R-module, N an R-submodule of M and I an ideal of R such that I ⊆ ann(M ).Then M is a 2-absorbing comultiplication R-module if and only if M is 2-absorbing comultiplication as an R/I-module.(b) Let R and R be any commutative rings, g : R → R a surjective homomorphism and M an R -module.If M is a 2-absorbing comultiplication R -module, then M is a 2-absorbing comultiplication R-module.Proof.(a) It is easy to see that N is a 2-absorbing R-submodule of M if and only if N is 2-absorbing submodule of M as an R/I-module.Now the assertion follows the fact that (0 : M J) = (0 : M (I + J)/I) for every ideal J of R. (b) Clearly, if N is a 2-absorbing R-submodule of M , then it is a 2-absorbing R -submodule of M .Assume that M is a 2-absorbing comultiplication R -module and let N be a 2-absorbing R-submodule of M .Then N = (0 : M J), where J = (0 : R N ); so I = g −1 (J) is an ideal of R with g(I) = J.It is enough to show that (0 : M J) = (0 : M I).Let m ∈ (0 : M J).If r ∈ I, then g(r) ∈ J, so g(r)m = 0. Thus rm = 0 for every r ∈ I; hence m ∈ (0 : M I).For the reverse inclusion, assume that x ∈ (0 : M I) and s ∈ J. Then s = g(a) for some a ∈ I.It follows that sx = g(a)x = ax = 0 for every s ∈ J; hence x ∈ (0 : M J), and we have equality.Assume that M is a 2-absorbing comultiplication module over a commutative ring R and let N be a non-zero pure submodule of M .Then the following hold:

Case 1 :Case 2 :
the field of fractions of R. Proof.By [10, Proposition 1.3], these modules are indecomposable.They are 2-absorbing comultiplication by Remark 2.5.It remains to be shown that there are no more indecomposable 2-absorbing comultiplication modules.Let M be an indecomposable 2-absorbing comultiplication and choose a non-zero element a ∈ M .Consider the annihilator, ann R (a) = {r ∈ R : ra = 0}, and the height h(a) = sup{n : a ∈ P n M } (so h(a) is a non-negative integer or ∞).If ann R (a) = P m+1 , then ann R (ap m ) = P .So, replacing a if necessary, it may be supposed that ann R (a) is 0 or P .Now we consider the various possibilities for h(a) and ann R (a).If abSpec(M ) = ∅, then Spec(M ) ⊆ abSpec(M ) gives M is a torsion divisible R-module with P M = M and M is not finitely generated by [27, Lemma 1.3, Proposition 1.4].We may assume that (0 : a) = P .By an argument like that in [11, Proposition 2.7 Case 2], M ∼ = E(R/P ).So we may assume that abSpec(M ) = ∅.If h(a) = n, then ann R (a) = P .Assume to the contrary, ann R (a) = 0. Say a = p n b.Then rb = 0 implies ra = 0 and so r = 0. Thus Rb ∼ = R.We also have that Rb is pure in M (see [13, Theorem 2.12 Case 1]).As M is a torsion-free R-module by [22, Theorem 10], we must have Rb is a prime submodule of M (see Remark 1.2 (b)) (so 2-absorbing submodule); hence R ∼ = Rb = (0 : M 0) = M , which is a contradiction by Remark 2.5 (a).So we may assume that h(a) = n, (0 : a) = P .Say a = p n b.Then we have Rb ∼ = R/P n+1 .Furthermore, Rb is pure in M .Hence, since Rb is a pure submodule of bonded order of M , we deduce that Rb is a direct summand of M by [22, Theorem 5]; hence M = Rb ∼ = R/P n+1 .Case 3: h(a) = ∞, ann R (a) = P .By an argument like that [11, Theorem 2.12, Case 4], we get that M ∼ = E(R/P ).Hence abSpec(M ) = ∅ by Remark 2.5 that is a contradiction.

( i )
If T is a 2-absorbing submodule of S, then T 1 is a 2-absorbing submodule of S 1 and T 2 is a 2-absorbing submodule of S 2 .(ii)abSpec(S) = ∅ if and only if abSpec(S i ) = ∅ for i = 1, 2.

Remark 3 . 2 .
Let R be the pullback ring as in(1), and let T be an R-submodule of a separated module S = (S 1 f1 −→ S f2 ←− S 2 ), with projection maps π i : S S i .

Proposition 4 . 5 .Corollary 4 . 6 .
Moreover, we can define a mapping π 1 = π 1 |T : T T 1 by sending (t 1 , t 2 ) to t 1 .Hence T 1 ∼ = T /(0 ⊕ Ker(f 2 ) ∩ T ) ∼ = T /(T ∩ P 2 S) ∼ = (T + P 2 S)/P 2 S ⊆ S/P 2 S. Let R be a pullback ring as in(1), and let M be an indecomposable 2-absorbing comultiplication non-separated R-module with M/P M finitedimensional top over R. If 0 → K → S → M → 0 is a separated representation of M , then S has finite-dimensional top and is pure-injective.Proof.Since S/P S ∼ = M/P M by [10, Proposition 2.6 (i)], we find that S has finite-dimensional top.Now the assertion follows from Theorem 4.4 and Corollary 3.7.Let R be a pullback ring as in (1) and let M be an indecomposable 2-absorbing comultiplication non-separated R-module with M/P M finite-dimensional over R. Consider the separated representation 0 → K → S → M → 0. By Proposition 4.3, S is pure-injective.So in the proofs of [10, Lemma 3.1, Proposition 3.2 and Proposition 3.4] (here the pure-injectivity of M implies the pure-injectivity of S by [10, Proposition 2.6 (ii)]) we can replace the statement "M is an indecomposable pure-injective non-separated R-module" by "M is an indecomposable 2-absorbing comultiplication 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 result: Let R be a pullback ring as in (1) and let M be an indecomposable 2absorbing comultiplication non-separated R-module with M/P M finite-dimensional over R, and let 0 → K → S → M → 0 be a separated representation of M .Then the following hold:(i) The quotient fields Q(R 1 ) and Q(R 2 ) of R 1 and R 2 do not occur among the direct summands of S.

able 2 -Theorem 4 . 7 .
absorbing comultiplication with M finite-dimensional top then S is a direct sum of just finitely many indecomposable separated 2-absorbing comultiplication modules and these are known by Theorem 3.6.In any separated representation 0 → K i −→ S ϕ −→ M → 0 the kernel of the map ϕ to M is annihilated by P , hence R-modules.As a consequence, any non-zero indecomposable 2-absorbing comultiplication separated module with 1-dimensional socle may occur only at one of the ends of the amalgamation chain (see [10, Proposition 3.4]).It remains to show that the modules obtained by these amalgamations are, indeed, indecomposable 2-absorbing comultiplication.We do that now and thus complete the classification of the indecomposable 2-absorbing comultiplication non-separated modules with finite-dimensional top.Let R = (R 1 → R ← R 2 ) be the pullback of two discrete valuation domains R 1 , R 2 with common factor field R. Then the indecomposable nonseparated 2-absorbing comultiplication modules with finite-dimensional top, up to isomorphism, are the following: (i) M = E(R/P ), the injective hull of R/P ; (ii) The indecomposable modules of finite length (apart from R/P which is separated), that is, M = s i=1 Ra i with p ns 1 a s = 0 = p m1 2 a 1 , p ni−1