Generalizations of S- Prime Ideals

Abstract

Let R be a commutative ring with identity and S be a multiplicative subset of R . In this paper we introduce the concept of almost S-prime ideal as a new generalization of S−prime ideal. Let P be a proper ideal of R disjoint with S. Then P is said to be almost S- prime ideal if there exists s ∈ S such that, for all x, y ∈ R if xy ∈ P − P 2 then sx ∈ P or sy ∈ P. Number of results concerning this concept and examples are given. Furthermore, we investigate an almost S- prime ideals of trivial ring extensions and amalgamation rings..