Relative Gorenstein Dimensions over Triangular Matrix Rings

Driss Bennis, Rachid El Maaouy, Juan Ramón García Rozas and Luis Oyonarte
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Driss Bennis: CeReMaR Research Center, Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, Rabat 10000, Morocco
Rachid El Maaouy: CeReMaR Research Center, Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, Rabat 10000, Morocco
Juan Ramón García Rozas: Department of Mathematics, University of Almería, 04071 Almería, Spain
Luis Oyonarte: Department of Mathematics, University of Almería, 04071 Almería, Spain

Mathematics, 2021, vol. 9, issue 21, 1-28

Abstract: Let A and B be rings, U a ( B , A ) -bimodule, and T = A 0 U B the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B . We show that when U is relative (weakly) compatible, we are able to describe the structure of G C -projective modules over T . As an application, we study when a morphism in T -Mod is a special G C P ( T ) -precover and when the class G C P ( T ) is a special precovering class. In addition, we study the relative global dimension of T . In some cases, we show that it can be computed from the relative global dimensions of A and B . We end the paper with a counterexample to a result that characterizes when a T -module has a finite projective dimension.

Keywords: triangular matrix ring; weakly Wakamatsu tilting modules; relative Gorenstein dimensions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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