RELATIVE SUBCOPURE-INJECTIVE MODULES

In this paper, copure-injective modules are examined from an alternative perspective. For two modules A and B, A is called B-subcopureinjective if for every copure monomorphism f : B → C and homomorphism g : B → A, there exists a homomorphism h : C → A such that hf = g. The class CPI−1(A) ={B : A is B-subcopure-injective} is called the subcopureinjectivity domain of A. We obtain characterizations of copure-injective modules, right CDS rings and right V-rings with the help of subcopure-injectivity domains. Since subcopure-injectivity domains clearly contains all copureinjective modules, studying the notion of modules which are subcopure-injective only with respect to the class of copure-injective modules is reasonable. We refer to these modules as sc-indigent. We studied the properties of subcopureinjectivity domains and of sc-indigent modules and investigated these modules over some certain rings.


Introduction and preliminaries
Throughout this paper, R will denote an associative ring with identity, and modules will be unital right R-modules, unless otherwise stated. As usual, the category of right R-modules is denoted by M od R.
Some new studies in module theory have focused on to approach to the injectivity from the point of relative notions. The injectivity domain In 1 (A) for a module A, is the class of all modules B such that A is B-injective [1]. Given A and B modules, A is called B-subinjective if for every monomorphism f : B ! C and homomorphism g : B ! A, there exists a homomorphism h : C ! A such that hf = g. Instead of using the injectivity domain, in latest articles, authors have proposed to consider an alternative sight so-called subinjectivity domain In 1 (A), contains of modules B such that A is B-subinjective ( [2]). It is clear that injectivity of A is equivalent to that In 1 (A) = M od R. If B is injective, then A is exactly Bsubinjective. So by [2,Proposition 2.3], the class of injective modules is the smallest studied and investigated over some certain rings, but we do not know whether copi-poor modules exist over arbitrary rings (see [15]).
Inspired by the notion of pure-subinjectivity from [11], in this paper we initiate the study of an alternative perspective on the analysis of the copure-injectivity of a module, as we introduce the notions of relative subcopure-injectivity and assign to every module its subcopure-injectivity domain. The aim of this paper is to investigate the viability of obtaining valuable information about a ring R from the perspective of subcopure-injectivity domain.
In Section 2, relative subcopure-injectivity and subcopure-injectivity domains of modules introduced. We investigate the properties of the notion of subcopureinjectivity and we compare subcopure-injectivity domains with (copure-)injectivity domains. We obtain characterizations of copure-injective modules, right CDS rings and right V-rings with the help of subcopure-injectivity domains.
In section 3, we introduced and studied the concept of cc-injective modules in terms of relative subcopure-injective modules. We give examples of cc-injective modules and compare cc-injective modules with cotorsion modules in Example 19. We prove that R is a right V-ring if and only if every cc-injective right R-module is injective. We investigate when the class of B-subcopure-injective modules is closed under extensions.
An R-module is copure-injective if and only if its subcopure-injectivity domain consists of M od R. Since subcopure-injectivity domains clearly contain all copureinjective modules, it is reasonable to investigate modules which are subcopureinjective only with respect to the class of copure-injective modules. It is thus to keep in line with [11], we refer to these modules as sc-indigent. In Section 4 of this paper, we studied and investigated sc-indigent modules over some certain rings. We compared sc-indigent modules with indigent modules and ps-poor modules.

Relative subcopure-injective modules
In this section, we study the B-subcopure-injective modules for a module B and examine its fundamental properties. De…nition 1. For two modules A and B, A is called B-subcopure-injective if for every copure monomorphism f : B ! C and homomorphism g : B ! A, there exists a homomorphism h : C ! A such that hf = g. The class CPI 1 (A) ={B : A is B-subcopure-injective} is called the subcopure-injectivity domain of A.
Hiremath proved in [8,Theorem 7] that every module can be embedded as a copure submodule in a direct product of co…nitely related modules. By [8,Proposition 3], every co…nitely related module is copure-injective and every direct product of copure-injective modules is copure-injective. This gives the below result that we use frequently in the sequel.
Lemma 2. For every module A, there exists a copure monomorphism : A ! C with C is copure-injective.
Our next Lemma gives a characterization of the B-subcopure-injective modules for a module B.
Lemma 3. Let A and B be two modules. The following conditions are equivalent: (1) A is B-subcopure-injective.
(2) For every homomorphism g : B ! A and every copure monomorphism (3) For every homomorphism g : B ! A and every copure monomorphism : B ! C with C direct product of co…nitely related modules, there exists h : C ! A such that h = g. (4) For every g : B ! A there exist a copure monomorphism : B ! C with C copure-injective and h : C ! A such that h = g. Proof.
(1) ) (2) Obvious. (3) ) (4) Let g : B ! A be a homomorphism. By Lemma 2, there exists a copure monomorphism : (4) ) (1) Let g : B ! A be a homomorphism and : B ! D a copure monomorphism. By (4), there exists a monic copure map : B ! C with C copure-injective and a homomorphism h : C ! A such that h = g. So by the copure-injectivity of C, there exists a homomorphism h : D ! C such that = h . Then h h : D ! A and h h = h = g. Hence, A is B-subcopure-injective. (1) A is copure-injective.
Proof. (1) ) (2) For any R-module B and any copure-injective module A, every copure monomorphism : B ! D and a homomorphism g : B ! A, there exists a homomorphism h : D ! A such that h = g. Hence, A is B-subcopure-injective and so B 2 CPI 1 (A). Consequently, CPI 1 (A) = M od R.
For any copure monomorphism : A ! B with B copure-injective and 1 A : A ! A, there exists a homomorphism g : B ! A such that g = 1 A . Thus splits. This means that A is copure-injective.
The next result asserts that subcopure-injectivity domain CPI 1 (A) of A how small can be. It should contain the copure-injective modules at least.
Proof. Suppose that each R-module is B-subcopure-injective for an R-module B. Then, by Proposition 4, B is copure-injective. Conversely, let A be any R-module and B a copure-injective module. Let g : B ! A be a homomorphism and : B ! C a copure monomorphism. Since B is copure-injective, the splitting map : B ! C gives the homomorphism : Clearly, CPI 1 (A) contains In 1 (A) for any module A. The following example shows that equality need not hold.
It is natural to investigate conditions to get the coincidence of the injectivity, and subcopure-injectivity domains, either for a certain class of modules or all the modules in M od R. We start by proving that, for all modules, subcopure-injectivity domains are the same as their subinjectivity domains over a right V-ring. Recall that a ring R is a right V-ring if and only if all exact sequences in M od R are copure if and only if all copure-injective modules are injective (see [8,Proposition 5]).
Corollary 7. Let R be a ring. The following conditions are equivalent: (1) R is a right V-ring.
Proof. (1) ) (2) It is easy since for any module A, over a right V-ring its extension is copure.
. This says that A is injective, and so R is a right V-ring by [8,Proposition 5].
Proposition 8. Let A be a module. The following conditions are equivalent: (2) ) (1) For a copure-injective extension C of A, C 2 CPI 1 (A), so A is also in CPI 1 (A) by (2). Then by Proposition 4, A is copure-injective.
The rings for which every right R-module is copure-injective are called right CDS, [8,Corollary 18]. As a result of Proposition 8, we get the following Corollary.
Corollary 9. Let R be a ring. The following conditions are equivalent: (1) R is right CDS.
Since R is a right CDS ring, A is copureinjective. The rest follows from Proposition 8.
(3) ) (1) For any right R-module A, CPI 1 (A) CPI 1 (A) by the hypothesis. Thus every right R-module A is copure-injective by Proposition 8, whence R is right CDS.
Remark 10. If A is R-subcopure-injective, for a ring R and a module A, then CPI 1 (A) and M od R need not be equal. For example if R is copure-injective ring that is not CDS, then for every module A, A is R-subcopure-injective by Proposition 5. But by the de…nition of right CDS ring, we can …nd a module A that is not copure-injective.
Proposition 11. Let A be a module. The following conditions are equivalent: (1) A is injective.
(1) ) (2) Let A be an R-module. Since R is semisimple, A is injective. The rest follows from Proposition 11.
(3) ) (1) For any right R-module A, CPI 1 (A) In 1 (A) by the hypothesis. Thus every right R-module A is injective by Proposition 11, whence R is semisimple.
In general, factors of copure-injective modules need not be copure-injective (see, [8,Remark 24]). But if R is a Dedekind domain, every copure factor of copureinjective module is copure-injective by [8,Corollary 28]. Hence, by the following Proposition, CPI 1 (A) is closed under copure homomorphic images over Dedekind domains for a module A. . Thus extends f . Then by the hypothesis, D C is copure-injective, so by Lemma 3, B C 2 CPI 1 (A).
i 2 I and f : B ! A i be a homomorphism. Then there exists a homomorphism g : C ! Q i2I A i such that g = i Ai f , where : B ! C is the monic map with C copure-injective and i Ai : Corollary 15. Let B be a module. Then B-subcopure-injective modules are closed under direct summands and …nite direct sums.

Proof. Let A be a module with decomposition
. Now the result follows.
The following shows that Proposition 14 do not hold for in…nite direct sums. Proof. Suppose C is a direct summand of B, and let f : C ! A be a homomorphism. By Lemma 2, there exist copure monomorphisms i : B ! D and j : C ! E with D and E copure-injective. Consider the following diagram: where i C : C ! B the inclusion map. Since D is copure-injective, there exists h : E ! D such that hj = ii C . Let C : B ! C be the projection map. Since A is B-subcopure-injective, there exists a homomorphism g : D ! A such that gi = f C . Then, (gh)j = g(hj) = gii C = f C i C = f , and so by Lemma 3, A is C-subcopure-injective.

cc-injective modules
In this section, we introduced and studied the concept of cc-injective modules in terms of relative subcopure-injective modules.
A module C is said to be co-absolutely co-pure (c.c. in short) if every exact sequence of modules ending with C is copure, equivalently Ext 1 R (C; A) = 0 for every co-…nitely related module A. Clearly every projective module is c.c. But the converse need not be true, for instance, the additive group Q is a c.c. Z-module but Q is not projective as a Z-module (see, [9, Example on page 290]).
Recall that a module A is called cotorsion if Ext 1 R (B; A) = 0 for every ‡at module B. A module A is called linearly compact if any family of cosets having the …nite intersection property has a nonempty intersection. A commutative ring is called classical if the injective hull E(S) of all simple modules S are linearly compact (see [17, §3]).
(2) By [9, Remark 15], c.c. modules need not be ‡at in general. By [9,Corollary 14] c.c. modules are ‡at over a commutative ring. So, in this case every cotorsion module is cc-injective.
Remark 20. Over a commutative ring R every simple R-module is cotorsion by [13,Lemma 2.14]. So by Example 19(2), every simple R-module is cc-injective.
Proof. Let A be a copure-injective module and B a c.c. module. By [9, Proposition 5], there exists a copure exact sequence 0 ! D ! P ! B ! 0 with P projective. If we apply Hom( ; A) to this sequence, we have Hom(P; A) ! Hom(D; A) ! Ext 1 R (B; A) ! Ext 1 R (P; A) = 0. Since A is copure-injective, Hom(P; A) ! Hom(D; A) is epic, and so Ext 1 R (B; A) = 0 for any c.c. module B. Hence A is cc-injective.
Proposition 22. For a ring R, the following conditions are equivalent: (1) R is a right V-ring.
(1) ) (3) Let A be a cc-injective R-module and B any R-module. Since R is right V , B is a c.c. module by [9,Proposition 4]. Thus Ext 1 R (B; A) = 0 for any R-module B, and so A is injective.
Proposition 23. Let B be an R-module and : B ! C a copure monomorphism with C copure-injective. If C=im( ) is c.c., then every cc-injective module is Bsubcopure-injective. Theorem 24. Let A and B be two modules. Consider the following conditions:

Proof. Let
(2) For every homomorphism g : B ! A, there exist a monomorphism : B ! C with C copure-injective and a homomorphism h : C ! A such that h = g. with C=B is c.c., there exists h : C ! A such that h = g.
(2) ) (1) Let : B ! C be a copure-monomorphism and g : B ! A a homomorphism. By (2), exists a monomorphism : B ! D with D copureinjective and a homomorphism h : D ! A such that h = g. Since D is copureinjective, there exists a homomorphism f : C ! D such that f = . Hence, (hf ) = h = g, and so (1) follows.
(3) ) (4) Let C be an extension of B with C=B is c.c. and g : B ! A a homomorphism. So, 0 ! B ! C ! C=B ! 0 is copure exact. Then consider the exact sequence with E cc-injective: is surjective, by (3), there exists a monomorphism f : B ! E and a homomorphism h : E ! A such that hf = g. Since is surjective, there exists a homomorphism : C ! E such that = f . Hence, h( ) = hf = g, and so (4) follows.
By the hypothesis, A is B-subcopure-injective. Conversely, let 0 ! A 0 ! A ! A 00 ! 0 be an exact sequence with A 0 and A 00 B-subcopure-injective. Then by Lemma 3, for every map g : B ! A, there exists a map h : C ! A 00 such that g = h where : B ! C is the copure monomorphism with C copure-injective. If we consider the pullback diagram: there exists a homomorphism : B ! D such that f = g and = .
By hypothesis, D is B-subcopure-injective, so by Lemma 3, there exists a homomorphism h 0 : C ! D such that h 0 = . Thus, f h 0 = f = g and so, A is B-subcopureinjective by Lemma 3.
A ring R is said to be right co-noetherian if every homomorphic image of a …nitely embedded R-module is …nitely embedded, equivalently for each simple right R-module S the injective hull E(S) is Artinian (see [10,Theorem]). Over a commutative noetherian ring, the injective hull of each simple right R-module is Artinian by [14,Exercise 4.17]. Thus every commutative Noetherian ring is co-noetherian. In the following, for an ideal I, we deal with an R-module structure of an R=I-module.
Proposition 26. Let R be a right co-noetherian ring and f : R ! S a ring epimorphism. If A is cc-injective S-module, then A is cc-injective R-module.
Proof. Let A be a cc-injective S-module. Since f : R ! S is a ring epimorphism, S = R=I for some ideal I of R and so A can be considered as R=I-module. Let C be an extension of A by a c.c. module F as R-modules. Since F is c.c., the exact sequence 0 ! A ! C ! F ! 0 is copure. Then A \ CI = AI for each right ideal I by [7, proposition 16]. Since A is an R=I-module, A \ CI = AI = 0, and so Since C A R I = C A+CI is c.c. as an R=I-module, so the second exact sequence splits and so does the …rst. Hence Ext 1 R (F; A) = 0, and A is cc-injective R-module.

sc-indigent modules
Indigent (resp. ps-poor) modules were introduced and some results about them were obtained in [2] (resp. [11]). Proposition 5 says that subcopure-injectivity domain of any module A contains all copure-injective modules, so studying the notion of modules which are subcopure-injective only with respect to the class of copure-injective modules is reasonable. It is thus to keep in line with [2], we refer to these modules as subcopure-injectively indigent (sc-indigent for short). In this section, sc-indigent modules investigated over certain rings and compared these modules with indigent modules and ps-poor modules.
De…nition 27. A module A is said to be subcopure-injectively indigent (sc-indigent for short), if CPI 1 (A) consists of only copure-injective modules.
Remark 28. Let A be a module with decomposition A = B C. If B is sc-indigent, then so is A, by Proposition 14.
Proposition 29. For a ring R, the following conditions are equivalent: (1) R is right CDS.
(4) 0 is an sc-indigent R-module. (5) R has an sc-indigent module and every sc-indigent R-module is copureinjective. (6) R has an sc-indigent module and every factor of an sc-indigent R-module is sc-indigent. (7) R has an sc-indigent module and every summand of an sc-indigent Rmodule is sc-indigent.
(3) ) (1) Let C be a copure-injective sc-indigent module and A a module. Since C is A-subcopure-injective, A is copure-injective. Then R is a right CDS ring.
Proof. Let A be an R-module with the exact sequence 0 ! A ! C ! C=A ! 0, where A ! C is a copure extension of A with C is copure-injective. Consider the sequence 0 ! Hom(C=A; B) ! Hom(C; B) ! Hom(A; B) ! Ext 1 (C=A; B).