Gorenstein Dimensions over Some Rings of the Form $R \oplus C$

TL;DR

This paper explores the relationships between $C$-Gorenstein dimensions over rings with a semidualizing module and Gorenstein dimensions over their trivial extensions, generalizing previous results to new algebraic structures.

Contribution

It extends existing results on Gorenstein dimensions to a broader class of rings formed via retract diagrams, including amalgamated duplication and pseudocanonical covers.

Findings

Established connections between $C$-Gorenstein and Gorenstein dimensions in new ring constructions.

Generalized Holm and Jørgensen's results to rings with specific retract diagrams.

Provided examples illustrating the applicability of the generalized theory.

Abstract

Given a semidualizing module $C$ over a commutative noetherian ring, Holm and J\o{}rgensen investigate some connections between $C$-Gorenstein dimensions of an $R$-complex and Gorenstein dimensions of the same complex viewed as a complex over the "trivial extension" $R \ltimes C$. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D'Anna and Fontana's amalgamated duplication and Enescu's pseudocanonical cover.