Covers and preenvelopes by \(V\)-Gorenstein flat modules. (Q2925615)

Let \(R\) denote a left and right Noetherian ring and let \(V\) be a dualizing module for \(R\). Let \(\mathbf W\) (resp. \(\mathbf X\); resp. \(\mathbf U\)) denote the class of all modules of the form \(V\otimes_RP\) (resp. \(V\otimes_RF\); resp. \(\Hom_{R^{op}}(V,E)\)), where \(P\) (resp. \(F\); resp. \(E\)) is a projective left (resp. flat left; resp. injective right) \(R\)-module. A left \(R\)-module \(M\) is said to be \(V\)-Gorenstein flat if an exact sequence of \(R\)-modules \(\cdots\to X_1\to X_0\to X^0\to X^1\cdots\) in \(\mathbf X\) exists such that \(M=\text{Ker}(X^0\to X^1)\) and both \(U\otimes_R-\) and \(\Hom_R(W,-)\) leave the sequence exact for all \(U\in\mathbf U\) and \(W\in\mathbf W\). The concept of a \(V\)-Gorenstein flat module coincides with that of a Gorenstein flat module if \(R\) is a Gorenstein ring.NEWLINENEWLINE Characterizations of \(V\)-Gorenstein flat modules are found in terms of the Bass and Auslander classes. The category \(V\)-\(\mathbf{GF}\) of \(V\)-Gorenstein flat left \(R\)-modules is shown to be stable, i.e. \(V\)-\(\mathbf{GF}=V\)-\(\mathbf{GF}^2\), and it is closed under direct limits, pure submodules and pure quotient modules.NEWLINENEWLINE It is shown that the pair \((V\)-\(\mathbf{GF},V\)-\(\mathbf{GF}^\perp)\) forms a perfect hereditary cotorsion theory in \(B^\ell(R)\). This theorem implies that every module in the left Bass class \(B^\ell(R)\) has a \(V\)-Gorenstein flat cover. More generally, it is then proved that every left \(R\)-module has a \(V\)-Gorenstein flat cover and a \(V\)-Gorenstein flat preenvelope.NEWLINENEWLINE In the final section, right self-injective rings, left perfect rings and left QF-rings are characterized in terms of Gorenstein and \(V\)-Gorenstein rings.