Prime Length of Crossed Products

Abstract

In this paper we show that the prime length of a crossed product $R*G$, where $R$ is a right Noetherian ring and $G$ is a polycyclic-by-finite group, is bounded by the plinth length of $G$ and the prime length of $R$. We begin by considering prime ideals in group rings of finitely generated Abelian groups, and generalize a theorem of J. E. Roseblade. We then use the description of prime ideals in crossed products given by D. S. Passman to achieve the result.