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 ofM. 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 2P(P), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x 2 Por 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)-primesubmodules and (n − 1, n) − m-prime submodules coincide (n,m 2).