Abstract In this work we introduce the concept of approximately regular ring as generalizations of regular ring, and the sense of a Z- approximately regular module as generalizations of Z- regular module. We give many result about this concept.
An -module is extending if every submodule of is essential in a direct summand of . Following Clark, an -module is purely extending if every submodule of is essential in a pure submodule of . It is clear purely extending is generalization of extending modules. Following Birkenmeier and Tercan, an -module is Goldie extending if, for each submodule of , there is a direct summand D of such that . In this paper, we introduce and study class of modules which are proper generalization of both the purely extending modules and -extending modules. We call an -module is purely Goldie extending if, for each , there is a pure submodule P of such that . Many c
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Throughout this work we introduce the notion of Annihilator-closed submodules, and we give some basic properties of this concept. We also introduce a generalization for the Extending modules, namely Annihilator-extending modules. Some fundamental properties are presented as well as we discuss the relation between this concept and some other related concepts.
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The main goal of this paper is to introduce and study a new concept named d*-supplemented which can be considered as a generalization of W- supplemented modules and d-hollow module. Also, we introduce a d*-supplement submodule. Many relationships of d*-supplemented modules are studied. Especially, we give characterizations of d*-supplemented modules and relationship between this kind of modules and other kind modules for example every d-hollow (d-local) module is d*-supplemented and by an example we show that the converse is not true.
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In this paper, we give a comprehensive study of min (max)-CS modules such as a closed submodule of min-CS module is min-CS. Amongst other results we show that a direct summand of min (max)-CS module is min (max)-CS module. One of interested theorems in this paper is, if R is a nonsingular ring then R is a max-CS ring if and only if R is a min-CS ring.
Let R be associative ring with identity and M is a non- zero unitary left module over R. M is called M- hollow if every maximal submodule of M is small submodule of M. In this paper we study the properties of this kind of modules.
The -s-extending modules will be purpose of this paper, a module M is -s-extending if each submodule in M is essential in submodule has a supplement that is direct summand. Initially, we give relation between this concept with weakly supplement extending modules and -supplemented modules. In fact, we gives the following implications:
Lifting modules -supplemented modules -s-extending modules weakly supplement extending modules.
It is also we give examples show that, the converse of this result is not true. Moreover, we study when the converse of this result is true.
Let R be a commutative ring with identity and M be a unitary R- module. We shall say that M is a primary multiplication module if every primary submodule of M is a multiplication submodule of M. Some of the properties of this concept will be investigated. The main results of this paper are, for modules M and N, we have M N and HomR (M, N) are primary multiplications R-modules under certain assumptions.