On weakly S -prime ideals of commutative rings

Let R be a commutative ring with identity and S be a multiplicative subset of R . In this paper, we introduce the concept of weakly S -prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S . We say that P is a weakly S -prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R , if 0 ≠ ab ∈ P , then sa ∈ P or sb ∈ P . We show that weakly S -prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S -Noetherian rings and S -principal ideal rings.