M. Ebrahimpour - Department of Mathematics, Faculty of Sciences, Vali-e-Asr University of Rafsanjan , P.O.Box ۵۱۸, Rafsanjan, Iran
F. Mirzaee - Department of Mathematics, Faculty of Sciences, Shahid Bahonar University of Kerman, Kerman, Iran.

Abstract. Let m; n ≥ 2 be two positive integers, R a commutative ring with identity and M a unitary R-module. A proper submodule P of M is an (n 1; n)-Φm-prime ((n 1; n)-weakly prime) submodule if a1; : : : ; an 1 2 R and x 2 M together with a1 : : : an 1x 2 Pn(P : M)m 1P (0 ̸= a1 : : : an 1x 2 P) imply a1 : : : ai 1ai+1 : : : an 1x 2 P, for some i 2 f1; : : : ; n 1g or a1:::an 1 2 (P : M). In this paper we study these submodules. Some useful results and examples concerning these types of submodules are given.

Quasi-local ring, Weakly prime submodule, (n-1, n)-weakly prime submodule, ϕm-prime submodule