Let $(R, m)$ be a commutative noetherian local ring and let $Gamma$ be a finite group. It is proved that if $R$ admits a dualizing module, then the group ring $Rga$ has a dualizing bimodule as well. Moreover, it is shown that a finitely generated $Rga$-module $M$ has generalized Gorenstein dimension zero if and only if it has generalized Gorenstein dimension zero as an $R$-module.

This paper studies the vanishing of $Ext$ modules over group rings. Let $R$ be a commutative noetherian ring and $ga$ a group. We provide a criterion under which the vanishing of self extensions of a finitely generated $Rga$-module $M$ forces it to be projective. Using this result, it is shown that $Rga$ satisfies the Auslander-Reiten conjecture, whenever $R$ has finite global dimension and $ga...

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. of finite index) unit group $$\mathcal {U}(\mathbb {Z}G)$$ integral ring $$\mathbb {Z}G$$ a G. These constructions rely explicit units in and proofs main results make use description Wedderburn components rational algebra {Q}G$$ . The latter relies primitive central idempotents re...