Prime ideals of fixed rings

Abstract

Let S be a non-commutative associative ring with an identity element and G be a finite group of ring automorhphisms of S. By exploiting Morita Theory, a bijection between subsets of SpecS and spec(S^G ) is constructed. This Morita formulation is shown to be equivalent to a much more natural definition of the bijection, one in which the strong relationship between the rings S and S^G is clearly manifest. Indeed, the bijection is shown to have implications for a number of ring-theoretic properties of rings S and S^G . One such property being prime rank. A topological treatment of the bijection using quotient Zariski topologies yields a homeomorphism which exhibits the structural similarities between S and S ^G. The final section is devoted to a special case: charS - q, q prime, and G a q-group. In this case, it is shown that a prime ideal of the skew group ring S*G is uniquely determined by its intersection with R.