Document Type : Other

Authors

• M. Ebrahimpour ¹
• F. Mirzaee ²

¹ Department of Mathematics, Faculty of Sciences, Vali-e-Asr University of Rafsanjan , P.O.Box 518, Rafsanjan, Iran

² Department of Mathematics, Faculty of Sciences, Shahid Bahonar University of Kerman, Kerman, Iran.

Abstract

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.

Keywords

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