Relative homological rings and modules

TL;DR

This paper develops a relative homological theory for rings and modules over commutative Noetherian rings, introducing new classes such as -relative regular, complete intersection, and Gorenstein rings, extending classical results.

Contribution

It introduces a relative framework for homological properties in commutative algebra, generalizing classical concepts to -relative contexts.

Findings

Defined -relative regular, complete intersection, and Gorenstein rings and modules.

Extended classical homological results to the -relative setting.

Demonstrated interactions between these new classes and existing homological properties.

Abstract

The study of rings and modules with homological criteria is a cornerstone of commutative algebra. Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. In this paper, a relative analogue of the theory of homological rings and modules is developed. We introduce the notions of $\frak a$-relative regular, $\frak a$-relative complete intersection, and $\frak a$-relative Gorenstein rings and modules. We extend some classical results by demonstrating some interactions between these types of rings and modules.