Primitive localizations of group algebras of polycyclic-by-finite groups
Let \(K[G]\) denote the group algebra of the polycylic-by-finite group \(G\) over the field \(K\), and assume that \(K[G]\) is prime. Let \(S\) be a multiplicatively closed set of nonzero central elements of \(K[G]\) and, if \(K\) is absolute, assume that \(S\) contains an element not in \(K\). This paper obtains necessary and sufficient conditions for the localization \(K[G]_S\) to be a primitive ring. Specifically, the criterion is: (1) \(A=K\langle S,S^{-1}\rangle\) is a \(G\)-domain, (2) \((Q(ZK[G]):Q(A))\) is finite, and (3) \(J(K[G]_S)=0\), where the semiprimitivity condition (3) is needed only when \(G\) is Abelian-by-finite. It follows that \(K[G]_{ZK[G]}\) is a primitive ring provided that \(ZK[G]>K\) when \(K\) is absolute.
scientific article; zbMATH DE number 1295424