phi-Classical Prime Submodules

Abstract

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each m is an element of M and elements a, b is an element of R, abm is an element of N implies that am is an element of N or bm is an element of N. Let phi : S(M) -> S(M) U {empty set} be a function where S(M) is the set of all submodules of M. We introduce the concept of phi-classical prime submodules. A proper submodule N of M is a phi-classical prime submodule if whenever a, b is an element of R and m is an element of M with abm is an element of N\phi(N), then am is an element of N or bm is an element of N.