Cyclic quiver rings and polycyclic-by-finite group rings (Q2705829)

Let \(R\) be a left and right Noetherian ring. \(R\) is called Iwanaga-Gorenstein if both injective dimensions \(\dim_R({_RR})\) and \(\dim_R(R_R)\) are finite. The main result of this paper reads as follows. If \(G\) is a polycyclic-by-finite group and \(R\) is Iwanaga-Gorenstein, then so are the skew polynomial ring \(R[x;\sigma]\) and the group ring \(R[G]\). More precisely, the authors show that \(\dim_{R[G]}R[G]\leq\dim_RR+h\), where \(h\) is the Hirsch number of \(G\). Similar results are proven in the case of the path ring \(R\widetilde A_n\), where \(\widetilde A_n\) is the cyclic quiver on \(n+1\) vertices.