The concept of epiform modules is a dual of the notion of monoform modules. In this work we give some properties of this class of modules. Also, we give conditions under which every hollow (copolyform) module is epiform.
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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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.
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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.
Let M be an R-module. In this paper we introduce the concept of quasi-fully cancellation modules as a generalization of fully cancellation modules. We give the basic properties, several characterizations about this concept. Also, the direct sum and the localization of quasi-fully cancellation modules are studied.
Let M be an R-module, where R is commutative ring with unity. In this paper we study the behavior of strongly hollow and quasi hollow submodule in the class of strongly comultiplication modules. Beside this we give the relationships between strongly hollow and quasi hollow submodules with V-coprime, coprime, bi-hollow submodules.
In this paper, we introduce the concept of s.p-semisimple module. Let S be a semiradical property, we say that a module M is s.p - semisimple if for every submodule N of M, there exists a direct summand K of M such that K ≤ N and N / K has S. we prove that a module M is s.p - semisimple module if and only if for every submodule A of M, there exists a direct summand B of M such that A = B + C and C has S. Also, we prove that for a module M is s.p - semisimple if and only if for every submodule A of M, there exists an idempotent e ∊ End(M) such that e(M) ≤ A and (1- e)(A) has S.
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Let R be an associative ring with identity and M be unital non zero R-module. A
submodule N of a module M is called a δ-small submodule of M (briefly N << M )if
N+X=M for any proper submodule X of M with M/X singular, we have
X=M .
In this work,we study the modules which satisfies the ascending chain condition
(a. c. c.) and descending chain condition (d. c. c.) on this kind of submodules .Then
we generalize this conditions into the rings , in the last section we get same results
on δ- supplement submodules and we discuss some of these results on this types of
submodules.
Let R be a commutative ring with identity . In this paper we study the concepts of essentially quasi-invertible submodules and essentially quasi-Dedekind modules as a generalization of quasi-invertible submodules and quasi-Dedekind modules . Among the results that we obtain is the following : M is an essentially quasi-Dedekind module if and only if M is aK-nonsingular module,where a module M is K-nonsingular if, for each , Kerf ≤e M implies f = 0 .