PSEUDO-RC-INJECTIVE MODULES

5189 | P a g e October 0 2 , 2 0 1 5  PSEUDO-RC-INJECTIVE MODULES Mehdi Sadiq Abbas, Mahdi SalehNayef Department of Mathematics, College of Science, University of Al-Mustansiriyah, Baghdad, Iraq Department of Mathematics, College of education, University of Al-Mustansiriyah, Baghdad,Iraq ABSTRACT The main purpose of this work is to introduce and study the concept of pseudo-rc-injective module which is a proper generalization of rc-injective and pseudo-injective modules. Numerous properties and characterizations have been obtained. Some known results on pseudo-injective and rc-injective modules generalized to pseudo-rc-injective. Rationally extending modules and semisimple modules have been characterized in terms of pseud-rc-injective modules. We explain the relationships of pseudo-rc-injective with some notions such as Co-Hopfian, directly finite modules. Indexing terms/


INTRODUCTION
Throughout, represent an associative ring with identity and all R-modules are unitary right modules. Let and be two R-modules, is called pseudo --injective if for every submodule of , any Rmonomorphism : → can be extended to an R-homomorphism : → . An R-module is called pseudo-injective, if it is pseudo N-injective. A ring R is said to be pseudo-injective ring, if is pseudo-injective module (see [5] and [14]).
A submodule of an -module is called rationally closed in (denoted by K ≤ ) if has no proper rational extension in [1]. Clearly, every closed submodule is rationally closed submodule (and hence every direct summand is rationally closed), but the converse may not be true (see [1], [6], [9]). M. S. Abbas and M. S. Nayef in [3] introduce the concept of rc-injectivity. Let 1 and 2 be -modules. Then 2 is called 1 --injective if every -homomorphism : → 2 , where is rationally closed submodule of 1 , can be extended to an -homomorphism : 1 → 2 . An -module is called -injective, if is --injective, for every -module . Anmodule is called -quasi-injective or self--injective, if is --injective.
In [15], an -module is called pseudo--c-injective if for any monomorphism from a closed submodule of to can be extended to homomorphism from in to . An R-module M is called rationally extending (or RCS-module), if each submodule of M is rational in a direct summand. This is equivalent to saying that every rationally closed submodule of M is direct summand. It is clear that every rationally extending R-module is extending [1]. An -module is said to be Hopfian (Co-Hopfian), if every surjective (injective) endomorphism ∶ → is an automorphism [16]. An -module is called directly finite if it is not isomorphic to a proper direct summand of [10]. An R-module M is said to be monoform, if each submodule of M is rational [17].

2Pseudo-rc-injectiveModules
We start with the following definition Definition 2.1Let and be two -modules .Then is pseudo -rationally closed-injective ( briefly pseudo -rcinjective ) if for every rationally closed submodule of , any -monomorphism : → can be extended to anhomomorphism : → . An -module is called pseudo-rc-injective, if is pseudo -rc-injective. ring is called self pseudo-rc-injective if it is a pseudo--rc-injective. (1)Every pseudo-injective module is rc-pseudo-injective. The converse may not be true in general, as following example let = -module. Then, clearly is rc-pseudo-injective, but Z is not pseudo-injective module. This shows that pseudo-rc-injective modules are a proper generalization of pseudo-injective.

Remarks 2.2
(2) Clearly every rc-injective is pseudo-rc-injective. The converse may not be true in general. For example, [7, lemma 2], let M be an R-module whose lattice of submodules is Where 1 is not isomorphic to 2, and the endomorphism rings of Ni are isomorphic to /2 where i=1,2. S. Jain and S. Singh in [7] are show that, is pseudo-injective (and hence by (1), is pseudo-rc-injective) which is not rc-quasiinjective, since 1 ⊕ 2 is rationally closed submodule of and the natural projection of 1 ⊕ 2 onto (i=1,2) can not be extended to an -endomorphism of , [7]. Therefore, M is not rc-injective module. This shows that pseud-rc-injective modules are a proper generalization of rc-injective modules.
(3) Obviously, every pseudo--rc-injective is pseudo -c-injective. The converse is not true in general. For example, consider the two -module = /9 and = /3 it is clear that is pseudo --injective but is not pseudo -rc-injective. This shows that pseud-rc-injective modules are stronger than of rc-injective modules.
In the following result we show that, for a uniform R-module the concepts of the rc-injective modules and pseudorc-injective are equivalents.

Theorem 2.3 Let be uniform -module. is a rc-injective if and only if is a pseudo-rc-injective module.
Proof:(⟹) Obviously.

(⟸)
Suppose that is a pseudo-rc-injective, let be rationally closed submodule of and ∶ → behomomorphism. Since is uniform module, either or − is a -monomorphism. First, if is -monomorphism, then by pseudo-rc-injectivity of , there exists -homomorphism ∶ → such that ∘ = .Finally, if − ismonomorphism, then by pseudo-rc-injectivity of , there exists ∶ → such that ∘ = − hence − = . Therefor is rc-injective. Proof: Follows from the definition and injections and projections associated with the direct product. □ The following corollary is immediately from proposition (2.5).
Corollary 2.6Let and be -modules where ∈ and is finite index set, if ⨁ =1 is pseudo -rc-injective, ∀ ∈ , then is pseudo--rc-injective. In particular every direct summand of pseudo-rc-injective R-module is pseudo-rcinjective. □ Proposition 2.7 Let and be -modules. If is pseudo --injective, then is pseudo --injective for every rationally closed submodule of .
Theorem 2.9If ⨁ is a pseudo-rc-injective module, then and are mutually rc-injective.

Corollary 2.11
For any positive integer n ≥ 2 , if is pseudo rc-injective , then is rc-quasi-injective. □ The following example shows that the direct sum of two pseudo-rc-injective is not pseudo-rc-injective in general. For a prim p, let 1 = and 2 = / , be a right -modules .Since 1 , and 2 are monoform then, 1 , and 2 are pseudo-rcinjective. But, by proposition (2.8), we have 1 ⨁ 2 is not pseudo-rc -injective module. Now, we consider the sufficient condition for a direct sum of two pseudo-rc-injective modules to be pseudo -rcinjective.

Theorem 2.12
The direct sum of any two pseudo-rc-injective modules is pseudo-rc-injective if and only if every pseudo -rcinjective module is injective. Recall that an -module is a multiplication if, each submodule of has the form for some ideal of [9].

Proposition 2.13 Every rationally closed submodule of multiplication pseudo-rc-injective -module is pseudo-rcinjective.
Proof: Let be a rationally closed submodule of a rationally closed submodule of and let : → be anmonomorphism. Since is a rationally closed of M. It follows that by [2,Lemma (3.2), is also a rationally closed submodule of M. Since is pseudo-rc-injective, then there exist an -homomorphism : → that extends . Since M is multiplication module, we have = for some ideal I of . Thus | = = = ≤ = . This show that is pseudo-rc-injective. □ In the following part we give characterizations of known R-modules in terms of pseudo-rc-injectivity.
We start with the following results which are given a characterization of rationally extending modules. Firstly, the following lemma is needed. Lemma 2.14 Let be rationally closed submodule of R-module . If is pseudo -rc-injective, then is a direct summand of . □ Proof: Since A is a pseudo M-rc-injective R-module, there exists an R-homomorphism : → . That extends The identity : → . Hence by [8,Corollary (3.4.10), = ⊕ , so that is a direct summand of .

Proposition 2.15
An -module is rationally extending if and only if every -module is pseudo --injective.

Proof:(⟹). It is similarly to prove [3, proposition (2.4).
(⟸). Follow from lemma (2.14). □ Note that, by proposition (2.15), every rationally extending -module is pseudo-rc-injective. But the converse is not true in general. As in the following example: consider the -module = / 2 where is prime number. It is clear that, is pseudo--injective (in fact, is rc-injective). Obviously, =< > is rationally closed submodule of but is not direct summand of . Thus is not rationally extending.
Proof: Let be an injective map belong to ( ) and I is identity R-homomorpism from to . By pseudo-rc-injectivity of , there exists an R-homomorphism : → such that ∘ = . Since is directly finite, we have ∘ = which is shows that is an R-automorphism. Therefore, is co-Hopfian. The other direction it is clear. □ The following corollary is immediately from proposition (2.17).

Corollary 2.18An rc-injective -module is directly finite if and only if it is Co-Hopfian. □
Since every indecomposable module is directly finite then by proposition (2.17), we obtain the following corollary.

Corollary 2.19
If is an indecomposable pseudo-rc-injective module then is a Co-Hopfian. □ In [33] was proved that every HopfianR-module is directly finite. Thus the following result follows from proposition (2.17).

Corollary 2.20
If is a pseudo-rc-injective and Hopfian -module .Then is a Co-Hopfian. □ For any an R-module M we consider the following definition.

Definition 2.21
An R-module M said to be complete rationally closed module (briefly CRC module), if each submodule of M is a rationally closed. It is clear that every semisimple module isCRC module, but the converse is not true in general.
For example 4 as Z-module is CRC module, but not semisimple since < 2 > is not direct summand of 4 .
An -module is said to be satisfies (C2)-condition, if for each submodule of which is isomorphic to a direct summand of , then it is a direct summand of [10].Recall that an R-module M is said to satisfy the generalized C2-condition (or GC2) if, any ≤ and ≅ , N is a summand of M [18].
Proof: Let be a pseudo-rc-injective CRC module, let ≤ and ≤ such that is isomorphic to with ≤ . Since is a pseudo-rc-injective then by corollary(2.6), we obtain is a pseudo--rc-injective. But ≅ thus, by remark (2.2)(9), is a pseudo -rc-injective. By assumption, we have K is rationally closed sub module of M. Thus, by Lemma (2.14), we get ≤ .Hence satisfiesC2. The last fact follows easily. □ Although the Z-module = is a pseudo-rc-injective, but it is not satisfies C2, since there is a submodule = (where ( ≥ 2)) of which is isomorphic to but it is not a direct summand in . This shows that the CRC property of the module in proposition (2.22) cannot be dropped.
In [4], an R-module is called direct-injective, if given any direct summand of , an injection map : → and every R-monomorphism : → , there is an R-endomorphism of such that = .
In [11,Theorem (7.13)], it was proved that, an R-module is a direct-injective if and only if is satisfies (C2)-condition. Thus by proposition (2.22) we can conclude the following result.

Proposition 2.23 Every pseudo-rc-injective CRC module is direct-injective. □
In [13, p.32], recall that a right -module is called divisible, if for each ∈ and for each ∈ which is not left zerodivisor, there exist ′ ∈ such that = ′ . In [4] was proved that every direct-injective -module is divisible. Thus we have the following corollary which follows from proposition (2.23). Corollary 2.24Every pseudo-rc-injective CRC module is divisible. October 0 2 , 2 0 1 5 Recall that an -module M is self-similar if, every submodule of is isomorphic to [12].The -module is both selfsimilar and pseudo-rc-injective module but it is not semisimple and CRC module.Also, 4 as -module is pseudo-rcinjective CRC module but it is not self-similar module.Note that from above examples the concepts CRC-modules and selfsimilar modules are completely different.
In the following result we show that the pseudo-rc-injective and semisimple -modules are equivalent under self-similar CRC modules. Theorem 2.25Let is a self-similar CRC module. Then the following statements are equivalent: is pseudo-rc-injective.
(ii)⟹(i). Let K be any submodule of M, then by self-similarity of M, we have K is isomorphic to M. Since M is pseudorc-injective CRC module thus, by proposition (2.22), M satisfy GC2-condition. So, K is a direct summand of M. therefore, is semisimple module. □