On weakly S-prime ideals of commutative rings

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 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.

Some results about weakly S-primary ideals of a commutative ring

“…(2) I(+)M is a weakly S(+)0-primary ideal of R(+)M . (2) =⇒ (1). Follows from Remark 2.10 since S(+)0 ⊆ S(+)M .…”

“…Note that if S consists of units of R, then the notions of S-primary and primary ideals coincide. In [1] F. A. A. Almahdi, E. M. Bouba and M. Tamekkante have defined a proper ideal P of R disjoint from a multiplicative subset S to be weakly S-prime if 0 ̸ = ab ∈ P implies sa ∈ P or sb ∈ P .…”

S-n-ideals of commutative rings

“…Many other generalizations of S-prime and S-primary ideals have been studied. For example, in [1], the authors defined I to be a weakly S-prime ideal if there exists an s ∈ S such that for all a, b ∈ R if 0 ̸ = ab ∈ I, then sa ∈ I or sb ∈ I. In 2015, Mohamadian [14] defined a new type of ideals called r-ideals.…”

On Weakly S-2-Absorbing Submodules

“…Of course a proper submodule P of M is called prime if am ∈ P for a ∈ R and m ∈ M implies a ∈ (P : R M) or m ∈ P where (P : R M) = {r ∈ R : r M ⊆ P }. Several generalizations of these concepts have been studied extensively by many authors [9], [13], [6], [16], [3], [11], [14], [5].…”

“…Here, for a multiplicatively closed subset S of R, they called a submodule P of an R-module M with (P : R M) ∩ S = ∅ a weakly S-prime submodule if there exists s ∈ S such that for a ∈ R and m ∈ M, if 0 = am ∈ P then either sa ∈ (P : R M) or sm ∈ P . In particular, a proper ideal I of R disjoint with S is said to be weakly S-prime if there exists s ∈ S such that for a, b ∈ R and 0 = ab ∈ I then either sa ∈ I or sb ∈ I [3].…”