Primitive Localizations of Group Algebras of Polycyclic-by-Finite Groups

Abstract

LetGbe a polycyclic-by-finite group such that Δ(G) is torsion-free abelian andKa field. Denote bySa multiplicatively closed set of non-zero central elements ofK[G]; ifKis an absolute field assume thatScontains an element not inK. Our main result is when the localizationK[G]Sis a primitive ring. This turns out to be equivalent to the following three conditions: (1)A=K〈S,S−1〉 is aG-domain, (2) (Q(ZK[G]):Q(A)) is finite, and (3)J(K[G]S)=0. In caseGis not abelian-by-finite, condition (3) is not needed. An immediate consequence is the following. LetKbe a field; in caseKis an absolute field assume that Δ(G)≠1. ThenK[G]ZK[G]is a primitive ring. In the final section a class of examples is constructed.