Throughout this thesis, S is a ring, G is a finite group of automorphisms of S and R is the fixed ring SG. We are concerned here with the correlation between properties of R and properties of S. In Chapter 2, we discuss certain finiteness conditions for the ring R. D.S. Passman has asked, "Is the fixed ring of kH, where k is a field and H is a polycyclic-by-finite group, Noetherian for any finite group G ?" We produce infinitely many examples for which the answer to this question is "yes". The most important results of this thesis are contained in Chapter 3. We develop the Morita prime correspondence of Chapter 1, §2, to produce results relating SpectR := (p ∈ SpecR: tr(S) ∉ p) to SpecfS := (P ∈ SpecS: Eg∈G g ∉ (P0*G)) where Po*G is an ideal in the skew group ring S*G. S. In Chapter 4, we restrict our attention to the case where S is a group algebra. We conclude this thesis in Chapter 5 with some results on localisation in the ring R. Many of these are inspired by the methods of Warfield in [W1]. We find that, with the necessary hypotheses, SpectR has the strong second layer condition. (Abstract shortened by ProQuest.)
