Quasi- essential submodules

Abstract

Let R be a commutative ring with identity, and M be a unitary R module. A proper submodule N of an R-module M is called an essential if ( ) 0 N K Ç ¹ for each non-zero submodule K of M, and a proper submodule L of an R-module M is called semi-essential if ( ) 0 L P Ç ¹ for each non-zero prime submodule P of M, which is a generalization of essential submodules. In this paper we give another generalization of essential submodule, we call it a quasi-essential submodulei, where we call a proper submodule H of an R-module M a quasi-essential if ( ) 0 H Q Ç ¹ for each non-zero quasi-prime R-submodule Q of M . Every essential submodule is a quasi-essential submodule ( semi-essential submodule), and every quasi-essential submodule is semi-essential but the converse is not true. Our main goal in this work is to study the basic properties of quasi-essential submodules, and semi-essential ideals in R. Also we study the modules that satisfies Acc(Dcc) on quasi-essential submodules.