Approximaitly Quasi-Prime Submodules and Some Related Concepts

Abstract

“Let  be a commutative ring with identity and  is a left unitary -module. A proper submodule  of  is called a quasi-prime submodule, if whenever , where ,  implies that either  or ”. As a generalization of a quasi-prime submodules, in this paper we introduce the concept of approximaitly quasi-prime submodules, where a proper submodule  of  is an approximaitly quasi-prime submodule, if whenever , where ,  implies that either  or , where  is the intersection of all essential submodules of . Many basic properties, characterization and examples of this concept are given. Furthermore, we study the behavior of approximaitly quasi-prime submodules under -homomorphisms. Finally, we introduced characterizations of approximaitly quasi-prime submodule in class of multiplication modules.