Perspectives on Gorenstein Homological Algebra

In the 1960s, Auslander and Bridger introduced the concept of G-dimension for finitely generated modules over a Noetherian ring. In 1995, Enochs and Jenda extended this concept to modules that are not necessarily finitely generated and defined what are now known as Gorenstein projective and injective modules, along with their homological dimensions. However, their investigation was limited to specific classes of rings. Later, in 2004, Holm provided new insights into Gorenstein projective and injective modules, as well as their homological dimensions, and extended several well-known results to rings that are not necessarily Noetherian. After that, more researchers became interested in studying this notion. A year later, Holm and Jørgensen introduced the concept of C-Gorenstein homological dimensions as a relative aspect of Gorenstein homological dimensions, where C is a semidualizing module. One of the motivations behind studying these modules is their relationship to the Gorenstein R-modules over R ⋉ C. In our work, we introduce the notion of relative W-Gorenstein modules, extending the classical Gorenstein projective modules in a relative setting. Our approach is based on the interplay between the additive closure of a given module and projectivity over its endomorphism ring.