S-Ideals: A Unified Framework for Ideal Structures via Multiplicatively Closed Subsets

In this paper, we study ideals defined with respect to arbitrary multiplicatively closed subsets S⊆R of a commutative ring R. An ideal I⊆R is called an S-ideal if for all a,b∈R, the condition ab∈I and a∈S implies b∈I. This is equivalent to the identity I=S⁻¹I∩R, where S⁻¹I is the extension of I in the ring of fractions S⁻¹R. The concept of S-ideals provides a unified framework encompassing several classical ideal types. For instance, r-ideals arise when S=reg(R), the set of regular elements. If S=R∖P for a prime ideal P, then the S-ideals coincide with P-primary ideals. Ideals that admit primary decomposition correspond to S-ideals for which S is the complement of a finite union of prime ideals. Moreover, z₀-ideals are S-ideals when S is the complement of a union of minimal prime ideals of R. We generalize several results known for r-ideals to this broader setting and investigate structural and closure properties of S-ideals in various contexts. As an application, we give a characterization of the von Neumann regularity of the localization S⁻¹R in terms of S-ideals. We also study the behavior of S-ideals in polynomial rings, idealizations, and amalgamated constructions with respect to different choices of S.