Mathematics (Oct 2021)

Relative Gorenstein Dimensions over Triangular Matrix Rings

  • Driss Bennis,
  • Rachid El Maaouy,
  • Juan Ramón García Rozas,
  • Luis Oyonarte

DOI
https://doi.org/10.3390/math9212676
Journal volume & issue
Vol. 9, no. 21
p. 2676

Abstract

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Let A and B be rings, U a (B,A)-bimodule, and T=A0UB 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 GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(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.

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