How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings

TL;DR

This paper develops methods to construct Gorenstein projective modules relative to duality pairs over Morita rings, extending previous results and applying to Ding projective modules.

Contribution

It generalizes the construction of duality pairs and Gorenstein projective modules over Morita rings, building on prior work on triangular matrix rings.

Findings

Constructed duality pairs of $ riangle$-modules from those of $A$- and $B$-modules.

Provided a method to build Gorenstein projective modules relative to duality pairs.

Applied the results to Ding projective modules.

Abstract

Let $\Delta =\left(\begin{smallmatrix} A & {_AN_B}\\ {_BM_A} & B \\\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_{A}N=0=N\otimes_{B}M$.We first study how to construct (complete) duality pairs of $\Delta$-modules using (complete) duality pairs of $A$-modules and $B$-modules, generalizing the result of Mao (Comm. Algebra, 2020, 12: 5296--5310) about the duality pairs over a triangular matrix ring. Moreover, we construct Gorenstein projective modules relative to complete duality pairs of $\Delta$-modules. Finally, we give an application to Ding projective modules.