Gorenstein (semi)hereditary rings with respect to a semidualizing module

TL;DR

This paper explores the properties of Gorenstein (semi)hereditary rings relative to a semidualizing module, introducing new classes of rings and establishing their coherence and semihereditary properties.

Contribution

It introduces and studies $C$-Gorenstein (semi)hereditary rings, extending Gorenstein homological concepts relative to a semidualizing module.

Findings

Every $C$-Gorenstein hereditary ring is coherent.

Such rings are also $C$-Gorenstein semihereditary.

Properties of finitely generated $G_C$-projective modules are characterized.

Abstract

Let $C$ be a semidualizing module. We first investigate the properties of finitely generated $G_C$-projective modules. Then, relative to $C$, we introduce and study the rings of Gorenstein (weak) global dimensions at most 1, which we call $C$-Gorenstein (semi)hereditary rings, and prove that every $C$-Gorenstein hereditary ring is both coherent and $C$-Gorenstein semihereditary.