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Relative Gorenstein Dimensions over Triangular Matrix Rings

Author

Listed:
  • Driss Bennis

    (CeReMaR Research Center, Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, Rabat 10000, Morocco
    These authors contributed equally to this work.)

  • Rachid El Maaouy

    (CeReMaR Research Center, Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, Rabat 10000, Morocco
    These authors contributed equally to this work.)

  • Juan Ramón García Rozas

    (Department of Mathematics, University of Almería, 04071 Almería, Spain
    These authors contributed equally to this work.)

  • Luis Oyonarte

    (Department of Mathematics, University of Almería, 04071 Almería, Spain
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Driss Bennis & Rachid El Maaouy & Juan Ramón García Rozas & Luis Oyonarte, 2021. "Relative Gorenstein Dimensions over Triangular Matrix Rings," Mathematics, MDPI, vol. 9(21), pages 1-28, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2676-:d:662200
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