New results related to a conjecture of Moore.
Let \(\Gamma'\) be a subgroup of a group \(\Gamma\) and let \(R\) be a ring with identity. We say that the triple \((\Gamma,\Gamma',R)\) satisfies Moore's condition if \(\Gamma'\) is of finite index in \(\Gamma\) and for all \(x\in(\Gamma-\Gamma')\) either there is an integer \(n\) such that \(1\neq x^n\in\Gamma'\) or the order of \(x\) is invertible in \(R\). Let \(R\Gamma\) be the group ring of the group \(\Gamma\) over the ring \(R\) and let \(M\) be an arbitrary \(R\Gamma\)-module, whose restriction to \(R\Gamma'\) is projective. If the triple \((\Gamma,\Gamma',R)\) satisfies Moore's condition, then the Moore conjecture asserts that \(M\) is also a projective \(R\Gamma\)-module [see \textit{L. G. Chouinard}, J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)]. In this paper the authors consider a similar analogue of this conjecture on injective modules. They show that the conjecture on injective modules is true for a very large class of groups. Moreover, they prove that the validity of the conjecture on injective modules implies the validity of it on projective and on flat modules. In addition, suppose that \(R\) is a commutative ring and \(M\) is an \(R\Gamma\)-module. Then it is proved that \(M\) is Gorenstein projective (resp. injective) if and only if \(M\) is Gorenstein projective (resp. injective) as an \(R\Gamma'\)-module [see \textit{E. E. Enochs} and \textit{O. M. G. Jenda}, Math. Z. 220, No. 4, 611-633 (1995; Zbl 0845.16005)].
scientific article; zbMATH DE number 6152308
scientific article