Strongly-Pseudo-Extending Modules and SPmodules

الملخص

Abstract

Through out this paper, R will be denoted an associative commutative ring with identity, and all R-modules are unitary (left) R-modules. An R-module M is called extending if every submodule of M is essential in a direct summand of M. Extending modules have been studied recently by several authors, among them M. Harada, B. Muller, P.F. Smith, and J. Clark [3].
In this work, we introduce and study in section one the concept of strongly-pseudo-extending module which is stronger property than extending module.
An R-module M is called strongly-pseudo-extending if, every submodule is essential in a pseudo stable direct summand of M. A non-zero submodule N of an R-module M is called pseudo stable if for each R-monomorphism [1]. And a non-zero submodule K of an R-module M is called essential in M, if for every non-zero submodule L of M [5].
Several characterizations of strongly-pseudo-extending modules are given. Moreover, we investigate direct decomposition for strongly-pseudo-extending modules. Also inherited property for strongly-pseudo-extending modules is studied. We show that a closed (and hence direct summand) submodules of strongly-pseudo-extending module are strongly-pseudo-extending.
In section two of this paper, as a proper generalization of fully-pseudo stable modules and as a link between extending modules, and strongly-pseudo extending modules, we introduce, and study the concept SP-module. An R-module is called Sp-module, if every direct summand of M is pseudo stable. Many examples, properties and characterizations of this concept are given; we assert that extending modules and strongly-pseudo-extending modules are linked by SP-module. Known modules related to SP-module are considered. A direct summand of SP-module is SP-module.