Gorenstein \(n\)-flat modules and their covers. (Q2925289)

\textit{S. B. Lee} introduced the concept of \(n\)-coherent rings and \(n\)-flat and \(n\)-absolutely pure modules [Commun. Algebra 30, No. 3, 1119-1126 (2002; Zbl 1022.16001)]. In this paper, the authors study the notion of Gorenstein \(n\)-flat modules (a left \(R\)-module \(M\) is Gorenstein \(n\)-flat if there exists an exact complex \(\mathbf F\) of \(n\)-flat modules with \(M=\mathrm{Ker}(F_0\to F_1)\) and such that \(E\otimes_R\mathbf F\) is exact whenever \(E\) is an \(n\)-absolutely pure right \(R\)-module) and they show that for right \(n\)-coherent rings the class of Gorenstein \(n\)-flat modules is closed under direct limits. The notion of Gorenstein \(n\)-absolutely pure module is also considered. For right \(n\)-coherent rings it is shown that \(M\) is a Gorenstein \(n\)-flat left \(R\)-module if and only if \(M^+=\Hom_{\mathbf Z}(M,\mathbf Q/\mathbf Z)\) is a Gorenstein \(n\)-absolutely pure right \(R\)-module. The last section of the paper is devoted to show that any left \(R\)-module over a right coherent \(n\)-coherent ring has a Gorenstein \(n\)-flat cover extending the result by \textit{E. E. Enochs} et al. [Math. Scand. 94, No. 1, 46-62 (2004; Zbl 1061.16003)]. In fact, the class of all Gorenstein \(n\)-flat \(R\)-modules is a Kaplansky class for any ring.