Title:

Skew group rings and maximal orders

In this thesis, we are concerned with the question of when a skew group ring of a finite group over a prime Noetherian ring is a maximal order. Contrary to what one might first expect, it is not necessary for the coefficient ring itself to be a maximal order; however, we do require that it be a Gmaximal order (where G is a group). Such objects are defined and discussed in Chapter 2. Chapter 3 is devoted to the proof of the main result of this work, In Chapter 4 we state and prove an important result which gives necessary and sufficient conditions for a crossed product to be local. Here, for an ideal p of a ring S, K(p) is the stabiliser of p in G. We then turn to Chapter 5, where we specialise Theorem 3.2.2 to give necessary and sufficient conditions for a skew group ring T of a finite group G over a commutative Noetherian domain S to be a maximal order. We continue our discussion, in Chapter 6, with a further result. This gives sufficient conditions for a skew group ring of a finite group over a prime Noetherian PI ring integral over its centre to be a tame order. If we allow the order of G to be a unit in the coefficient ring, then one of these conditions becomes redundant. Finally, in Chapter 7, we give sufficient conditions for a skew group ring of a finitely generated nilpotent group over a prime Noetherian ring to be a maximal order. COROLLARY 7.1.5 Let R be a prime Noetherian ring and G a finitely generated nilpotent group acting on R. Denote by H the torsion subgroup of G and suppose that the elements of H are Xouter on R. Suppose that the skew group ring R*H is a maximal order. Then the skew group ring R*G is a maximal order.
