M-Sp-Injective Modules

Abstract

In this paper we study M -small principally injective (in short, M -sp-injective) module which is the generalization of M -principally injective module. We prove that if M is finite dimensional and quasi-sp-injective then its endomorphism ring S is semi-local ring. We characterize the M -sp-injective module with the help of epi-retractable modules. Keywords: Small submodule; M -cyclic submodule; M -sp-injective modules; quasi-spinjective modules and epi-retractable module. AMS Subject Classification: 16D10, 16D50, 16D60 1250005-1 V. Kumar et al.N . A module M is called quasi-principally (or semi) injective, if it is M -principally injective. In this paper we introduce the notion of M -small principally injective modules and quasi-small principally injective modules which we abbreviate as Msp-injective and quasi-sp-injective modules. A submodule K of an R-module M is said to be small in M , written as K M , if for every submodule L ⊆ M withFor details see [1,5]. In this paper, we study M -sp-injective modules and M -sp-injective rings and also give an example of an M -sp-injective module which is not an M -principally injective module.Consider the following condition for an R-module M ., it is called continuous, if it satisfies (C 1 ) and (C 2 ), and quasi-continuous, if it satisfies (C 1 ) and (C 3 ). For undefined notations and terminologies see [1,14]. M-Small Principally Injective ModulesGeneralizing the notion of Sanh et al. [13], we now give a new definition. DefinitionAn R-module N is called M -small principally injective, if for every small M -cyclic submodule K of M , any homomorphism from K to N can be extended to a homomorphism from M to N . If M is a M -small principally injective module, then it is called a quasi-small principally injective module, or M -sp-injective module, and for short a ring R is called a right small-principally injective ring, if R R as a right R-module. Example(1) Every M -principally injective module is a M -sp-injective module.(2) Every semi-simple module is an M -sp-injective module.We now give an example of M -sp-injective modules which is not M -principally injective. ExampleZ is a Z-small principally injective module, but it is not Z-principally injective, because the only small Z-cyclic submodule of Z is 0.The following lemmas are the generalizations of [13, Lemmas 2.2-2.4]. Lemma 2.1. Let M i (1 ≤ i ≤ n) be M-sp-injective modules. Then n i=1 M i is M -sp-injective module. 1250005-2 M -SP-Injective Modules Proof. The proof is similar to that of [13, Lemma 2.2]. Lemma 2.2. Let X be an M-cyclic submodule of M . If X is M-sp-injective module, then it is a direct summand of M . Proof. The proof is similar to that of [13, Lemma 2.3]. Lemma 2.3. Every direct summand of M-sp-injective module is an M-sp-injective module. Proof. By the same argument as that given in [13, Lemma 2.4]. 1250005-5 V. Kumar et al. Proposition 2.6. For a hollow R-module M, the following conditions are equivalent :(1) M is quasi-sp-injective module;(2) M is quasi-principally injective module.Proof. It is ...

Pseudo - y - closed - Injective Modules

“…If D is D -injective, then D is said to be self--closed-injective or quasi--closed-injective (briefly, D is quasi--injective). We say D is -injective if it is K--injective, for any R-module K. In [7] and [8], an R-module D is a pseudo-K-c-injective if for any ∈ Mon R (A,D) where A ⊆c K, there exists ∈Hom R (K, D) such that = . An R-module D is said to be co-Hopfian (Hopfian) if each surjective (injective) endomorphism : D → D is automorphism see [9].…”

“…We say D is a fully stable if any submodule of D is stable see [10]. An homomorphism : B ⟶D is called C-homomorphism if (B) is closed in D see [8].…”

$S$-$M$-cyclic submodules and some applications

“…The notion of M -principally injective module has attracted many researchers and it has been studied in many papers. See, for examples, [8], [11], [12] and [14]. Recall from [5] that a right R-module N is called pseudo-M -principally injective, if every monomorphism from an M -cyclic submodule X of M to N can be extended to an R-homomorphism from M to N .…”