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).
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