Fixed rings and noncommutative invariant theory

Abstract

This chapter discusses fixed rings and the noncommutative invariant theory. The theory of fixed rings deals with the relationships between a given rings R acted on by a finite group G of its automorphisms and the fixed ring. It is easy to see that “t” is an R G -bimodule homomorphism from R to R G . The modern development of the theory is inspired by using non-commutative Galois Theory for skew fields–– the fixed ring of an Ore domain is an Ore domain. The minimal number of prime sub direct cofactors (the prime dimension) of the fixed ring is equal to the number s in this decomposition. In the classical Galois Theory a (commutative) field F is always a finite-dimensional space over fixed subfields F G (for finite G). This fact is still valid for skew fields and for simple Artinian rings A under the condition that there is no additive |G|-torsion. If A is a skew field then both the left and right dimensions are finite. Some kind of finiteness relations of the ring R with the fixed ring R G still exists. This kind of relations is the second important tool in the modem theory of fixed rings. The relation of fixed ring theory with polynomial identities (PI)-rings is also described.