A unified approach to various Gorenstein modules. (English) Zbl 1509.18011
Let \(R\) be a ring, and \(\mathcal{V}\), \(\mathcal{W}\), \(\mathcal{Y}\), and \(\mathcal{X}\) four classes of left \(R\)-modules. The authors introduce a notion of \((\mathcal{V},\mathcal{W},\mathcal{Y},\mathcal{X})\)-Gorenstein module which unifies a number of well-known Gorenstein modules such as Gorenstein projective (injective) modules, Ding projective (injective) modules, Gorenstein AC-projective (injective) modules, and Gorenstein C-projective (injective) modules. Under certain mild assumptions on the classes \(\mathcal{V}\), \(\mathcal{W}\), \(\mathcal{Y}\), and \(\mathcal{X}\), they first study the essential properties of \((\mathcal{V},\mathcal{W},\mathcal{Y},\mathcal{X})\)-Gorenstein modules, and then apply the obtained results to establish the Foxby equivalences associate to \((\mathcal{V},\mathcal{W},\mathcal{Y},\mathcal{X})\)-Gorenstein modules. Many results on Gorenstein modules scattered in the literature can then be obtained as corollaries of this general setting.
Reviewer: Hossein Faridian (Clemson)
MSC:
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 18B05 | Categories of sets, characterizations |
| 18G20 | Homological dimension (category-theoretic aspects) |
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