PRIME IDEALS IN CROSSED PRODUCTS OF FINITE-GROUPS

Let R * G be a crossed product of the finite group G over the ring R. In this paper we discuss the relationship between the prime ideals of R*G and the G-prime ideals of R. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime ideals P of R*G such that P∩R is a fixed G-prime ideal of R. As an application, we prove that if G is a finite group of automorphisms of R, then the prime (primitive) ranks of R and of the fixed ring R G are equal provided •G•-∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group. © 1979 Hebrew University.