Enochs, Edgar E.; Jenda, Overtoun M. G.; López-Ramos, J. A. A noncommutative generalization of Auslander’s last theorem. (English) Zbl 1103.16011 Int. J. Math. Math. Sci. 2005, No. 9, 1473-1480 (2005). Let \(R\) be a ring, a left \(R\)-module \(M\) is said to be Gorenstein projective if there exists an exact sequence \(\cdots P_1\to P_0\to P^0\to P^1\to\cdots\) of projective left \(R\)-modules which remains exact whenever \(\operatorname{Hom}_R(-,P)\) is applied to it for every projective module \(P\) and such that \(M=\ker(P^0\to P^1)\). The finitely generated Gorenstein projective modules coincide with the modules of \(G\)-dimension zero. These modules were introduced by M. Auslander [in Séminaire d’algèbre commutative 1966/67, École Normale Supérieure de Jeunes Filles, Paris (1967; Zbl 0157.08301)]. For Gorenstein commutative local rings, Auslander announced that every finitely generated module has a finitely generated Gorenstein projective cover (or equivalently, a minimal maximal Cohen-Macaulay approximation). The first two authors and J. Xu [in Algebr. Represent. Theory 2, No. 3, 259-268 (1999; Zbl 0938.13005)] showed the result for Cohen-Macaulay rings having a dualizing module. In this paper, the authors study the noncommutative situation. A ring \(R\) is said to be left \(n\)-perfect if every left flat \(R\)-module has projective dimension less than or equal to \(n\). It is shown that every finitely generated left \(R\)-module in the Auslander class over an \(n\)-perfect ring \(R\) having a dualizing module and admitting a Matlis dualizing module has a finitely generated Gorenstein projective cover. Reviewer: Blas Torrecillas (Almeria) Cited in 2 Documents MSC: 16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16D40 Free, projective, and flat modules and ideals in associative algebras 16E10 Homological dimension in associative algebras 16D20 Bimodules in associative algebras 16L30 Noncommutative local and semilocal rings, perfect rings Keywords:Gorenstein projective modules; Matlis dualizing modules; precovers; covers; \(n\)-perfect rings; dualizing bimodules; Auslander classes; Bass classes; projective dimension; Noetherian rings Citations:Zbl 0157.08301; Zbl 0938.13005 PDFBibTeX XMLCite \textit{E. E. Enochs} et al., Int. J. Math. Math. Sci. 2005, No. 9, 1473--1480 (2005; Zbl 1103.16011) Full Text: DOI EuDML