Generalized twisted group rings. (Q2583022)

Let \(R\) be a Dedekind domain, and let \(G\) be an arbitrary group. The authors consider generalized group rings \(R*G\), twisted by a generalized 2-cocycle \(\alpha\colon G\times G\to R\setminus\{0\}\), i.e. with values not necessarily in \(R^\times\). Then \(H:=\{ x\in G\mid\alpha(x,x^{-1})\in R^\times\}\) is a subgroup of \(G\). For any maximal ideal \(p\) of \(R\), the corresponding discrete valuation \(v_p\) gives rise to a 2-cocycle \(\alpha_p=v_p\alpha\) with values in \(\mathbb{Z}\). If \(\alpha_p\) has finite order in \(H^2(G,\mathbb{Z})\), then \(\alpha_p\) is equivalent to a cocycle \(m_p\) with values in \(\{0,1\}\). Assume this for all \(p\). Then, if for each \(x\in G\), some power \(x^k\) belongs to \(H\), the authors show that the order \(R*G\) is graded maximal and graded hereditary. In case \(R\) is a discrete valuation domain, and \(G\) is finitely generated and finite over its centre, the authors give sufficient conditions for \(R*G\) to be maximal over a central subring. For an arbitrary commutative ring \(R\), they also study the case when \(R*G\) is an Azumaya algebra.