# On Generalization of prime submodules

Document Type : Research Paper

Authors

1 Shahid Bahonar University Of Kerman

2 Shahid Bahonar University of Kerman

Abstract

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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Main Subjects

### History

• Receive Date: 29 June 2011
• Revise Date: 18 August 2012
• Accept Date: 18 August 2012
• First Publish Date: 15 October 2013