On 𝑆-weakly prime ideals of commutative rings

Abstract

Abstract In this paper, we introduce and study the notion of an 𝑆-weakly prime ideal which is a generalization of a weakly prime ideal. Let 𝐴 be a commutative ring and 𝑆 a multiplicative set of 𝐴. We define a proper ideal 𝑃 of 𝐴 to be 𝑆-weakly prime if there exists s ∈ S s\in S such that, for all a , b ∈ A a,b\in A with 0 ≠ a ⁢ b ∈ P 0\neq ab\in P , then s ⁢ a ∈ P sa\in P or s ⁢ b ∈ P sb\in P . We describe the behavior of an 𝑆-weakly prime property across various ring-theoretic constructions such as direct product, homomorphic image, localization, trivial ring extensions and amalgamation rings.