A connection between some properties of \(n\)-group rings and group rings (Q2724018)

An \(n\)-group ring \(RG\) is a \((2,n)\)-ring, every element of which can be uniquely represented as a finite sum \(\sum_i r_ig_i\) (\(r_i\in R\), \(g_i\in G\)), where \(R\) is an associative ring with unity and \(G\) is an \(n\)-group. Inverse elements of an \(n\)-group ring are defined and connections with inverse elements in a group ring are established. It is proved that the minimal condition for left ideals is satisfied in an \(n\)-group ring \(RG\) iff it is satisfied in the corresponding group ring \(RG_0\). Similar results are obtained for polynomial identities and for generalized nilpotent ideals. It is proved that if two \(n\)-group rings are isomorphic, then the corresponding group rings are also isomorphic.