\(V\)-Gorenstein projective, injective and flat modules. (Q1945740)

The main aim of the present paper is to study properties of \(V\)-Gorenstein projective, injective, respectively, flat modules. These results generalize some well known properties of projective, injective, respectively, flat modules. It is proved that the class of \(V\)-Gorenstein projective modules is closed with respect to kernels of epimorphisms (Theorem 2.3), direct sums and direct summands (Proposition 2.4), and if \(R\) is left perfect this class is closed with respect to direct limits of chains of \(V\)-Gorenstein projective modules. In Theorem 2.8 the authors provide a characterization for QF-rings using \(V\)-Gorenstein projective modules. Similar results are proved for \(V\)-Gorenstein injective, respectively, flat modules. For instance, it is proved that a left \(R\)-module \(M\) is \(V\)-Georenstein flat if and only if the dual \(M^+\) is \(V\)-Gorenstein injective (Theorem 3.3).