Relative homological rings and modules

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 $\mathfrak{a}$ 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 $\mathfrak{a}$-relative regular, $\mathfrak{a}$-relative complete intersection, and $\mathfrak{a}$-relative Gorenstein rings and modules. We extend some classical results by demonstrating some interactions between these types of rings and modules.

to appear in the Rocky Mountain Journal of Mathematics