Generalizations of prime submodules over non-commutative rings

Throughout this paper, R is an associative ring (not necessarily commutative) with identity and M is a right R-module with unitary. In this paper, we introduce a new concept of empty;-prime submodule over an associative ring with identity. Thus we define the concept as following: Assume that S(M) is the set of all submodules of M and Oslash; : S(M) ! S(M) [ f;g is a function. For every Y 2 S(M) and ideal I of R; a proper submodule X of M is called Oslash;-prime, if YI sube; X and YI nsub; Oslash;(X); then Y sube; X or I sube; (X :R M): Then we examine the properties of Oslash;-prime submodules and characterize it when M is a multiplication module.