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?1 I ? R, where S?1 I 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 containing P coincide with P-primary ideals.