Quasi-Coregular Modules

In this paper, we introduce the concept of quasi – copure submodules which is ageneralization of a copure submodules. Weused this concept to define the class of quasi – coregular module, where an R-module M iscalled quasi –coregular module if every submodule of M is quasi-co-pure. Many results about thisconcept are proved.


Introduction
Throughout this note, R is commutative ring with identity and M be a unitary R-module. A submodule N of an R-module M iscalled pure in M if IN = N ∩ IM for every ideal I of R [1]. An R-module M regular module if every sub module of M is pure [2] ."Ansari and F. Farshadifar"in [3] introduced the concept of copure submodules, where a submodule N of M is called copureif [N : I] = N + [0: I] foreach ideal I ofR. First recall that a submodule N of an R-module M iscalled a quasi -pure if foreach ‫ݔ‬ ∈ M and ‫ݔ‬ ∉ N, there exists apure submodule L of M such that N ⊆ L and ‫ݔ‬ ∉ L and anRmodule M iscalled quasi -regular1module ifevery submodule ofM is quasi -puree [4]. Thispaper iis structured in two sections. In section one-wegive new results about quasi-copure submodules. Insection two, westudy theconcept of quasi −coregular mmodules. We give some relationships between quasi −coregular modules (rings) and quasi − regular modules (rings).

Quasi ‫܍ܚܝܘ‪-Co‬‬ Submodules:
In this section we introduce the concept of quasi -copuressubmodule. Weinvestigate the basic properties of these type of submodules are analogous to the properties of copuressubmodules.
Definition (2.1): Let M bean R-module. A submodule N of M iscalled a quasi -copure submodule of M if foreach ‫ݔ‬ ∈ M and ‫ݔ‬ ∉ N, thereexists a copuressubmodule L ofM suchthat N ⊆ L and ‫ݔ‬ ∉ L.

MAICT
Recallthat anRR-module M iscalled copure simple if M and < 0 > aretthe only copure submodule of M [6]. Proof: Assume that N is a quasi -co-pure submodule of M. It clearthat N ∩ ࢻ L . We have to showthat ∩ ࢻ L N, let y ∈ ∩ ࢻ L , then y∈ L , foreach α, suppose y ∉ N. Since N is quasi -copure, hence y isnot contained in any copure submodule that contains N, which is acontradiction, thus y ∈ N. sothat ∩ ࢻ L N and N = ∩ ࢻ L .
Conversely assume that N =∩ ࢻ L , where L are a copure submodules of M foreach α, and containing N. Let ‫ݔ‬ ∈ M and ‫ݔ‬ ∉ N. Since N =∩ ࢻ L , then ∉ ∩ ࢻ L . Thus N L and ‫ݔ‬ ∉ L , forsome α. Hence N is quasi -copure.
Remark (2.8): Every direct summand of an R-module M is quasi -copure.
Proof: Since every direct summand of an R-module M is copure [5], and every copure is quasicopure, hence is quasi -copure.

Basic Results for Quasi -co-regular modules
Inthis section, we introduce and study the class of quasi -co regular modules. However, we give some basic results about this concept. Beside these we study the direct summand of quasi-co-regular modules and direct sum of quasi-co-regular modules.
Recall that an R-module M is called coregular if every submodule is co-pure and a ring R is coregular if every ideal of R is co-pure [5]. An R-module M is quasi-regular if every submodule of M is quasi-pure [4].
(2) Every semisimple R-module is coregular, hence is quasi -coregular. But the converse isnot true in general. We have no example.
(4) Z ସ as Z − module isnot quasicoregular . since not coregular [4]. , hence ‫ݔ‬ ∉ K. Since K is quasi -co-pure in M, then there exists a co-pure submodule L of M suchthat K ⊆ L and ‫ݔ‬ ∉ L, hence ‫̅ݔ‬ ∉ , since L is copure, then by [5] is copure. Also Definition (3.9): Let R be a ring, R is called a quasi -coregular ring ifevery ideal in R is quasicopure.
Remark and Example (3.10): (1) It is clearthat every coregular ring is quasi-coregular and we have no example of quasicoregular, which isnot coregular.