On Generalization of prime submodules

Let R be a commutative ring with identity and M be a unitary R-module. Let : S(M) −! S(M) [{;} be a function, where S(M) is the set of submodules of M. Suppose n  2 is a positive integer. A proper submodule P of M is called (n − 1, n) − -prime, if whenever a1,. .. , an−1 2 R and x 2 M and a1. .. an−1x 2 P(P), then there exists i 2 {1,. .. , n − 1} such that a1. .. ai−1ai+1. .. an−1x 2 P or a1. .. an−1 2 (P: M). In this paper we study (n − 1, n) − -prime submodules (n  2). A number of results concerning (n−1, n)−-prime submodules are given. Modules with the property that for some , every proper submodule is (n−1, n)−- prime, are characterized and we show that under some assumptions (n−1, n)-prime submodules and (n − 1, n) − m-prime submodules coincide (n,m  2).