On relative Gorenstein homological dimensions with respect to a dualizing module. (English) Zbl 1474.13031

Summary: Let \(R\) be a commutative Noetherian ring. The aim of this paper is studying the properties of relative Gorenstein modules with respect to a dualizing module. It is shown that every quotient of an injective module is \(G_C\)-injective, where \(C\) is a dualizing \(R\)-module with \(\mathrm{id}_R(C) \leq 1\). We also prove that if \(C\) is a dualizing module for a local integral domain, then every \(G_C\)-injective \(R\)-module is divisible. In addition, we give a characterization of dualizing modules via relative Gorenstein homological dimensions with respect to a semidualizing module.


13D05 Homological dimension and commutative rings
13D45 Local cohomology and commutative rings
18G20 Homological dimension (category-theoretic aspects)
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