I-Primal Submodules

Abstract

Abstract In module over commutative ring with identity, prime notions of submodule are growing vastly. Furthermore, there are also notions of primal submodules. Suppose N be a submodule of R-module M. An element r of R is called prime to N if rm ∈ N with m element of M implies m ∈ N. The submodule N is called primal if the set of elements which not prime to N forms an ideal. The generalization of elements which prime to N is proposed and referred as elements that almost prime to N. An element r ∈ R is called almost prime to N if rm ∈ N – (N: M)N for m element of M implies m is an element of N. In this article, we introduce the generalization of elements which are prime to N as elements which are I-prime to N by changing the set (N:M) into an arbitrary ideal of R. With this notion, we can generalize primality of submodule N to I-primality of submodule N. We called N the submodule of M is I-primal submodule if the set of elements which are I-prime to N forms an ideal of R. Furthermore, we give some characterizations of I-primal submodules and their relation to other prime and primal notions.