Prime Ideals in Polycyclic Crossed Products

Abstract

In this paper, we describe the prime ideals $P$ in crossed products $R \ast G$ with $R$ a right Noetherian ring and with $G$ a polycyclic-by-finite group. This is achieved through a series of reductions. To start with, we may assume that $P \cap R = 0$ so that $R$ is a $G$-prime ring. The first step uses a technique of M. Lorenz and the author to reduce to a prime ring and a subgroup of finite index in $G$. Next if $R$ is prime, then we show that the prime ideals of $R \ast G$ disjoint from $R$ are explicitly determined by the primes of a certain twisted group algebra of a normal subgroup of $G$. Finally the prime ideals in twisted group algebras of polycyclic-by-finite groups are studied by lifting the situation to ordinary group algebras where the results of J. E. Roseblade can be applied.