Applications of Jones index to rings with Artinian quotient rings (Q1330036)

The author's earlier results on the Jones index for inclusions of finite- dimensional algebras [J. Algebra 160, 461-491 (1993)] are extended to inclusions of rings possessing artinian classical quotient rings. Since the original Jones index is not preserved under localization, the author introduces a modification called the reduced Jones index, in which the use of minimal numbers of generators is replaced by reduced rank. The main applications are to the fixed ring of a finite group \(G\) of automorphisms of a polynomial identity ring \(R\), with \(| G|^{- 1} \in R\). For example, it is proved that given any prime ideal \(P\) in \(R\), the number of prime ideals of \(R^ G\) minimal over \(P \cap R^ G\) is bounded above by the reduced Jones index of \(R^ G/(P \cap R^ G)\) in \(R/P\). In the case that \(R\) is prime, it is shown that the reduced Jones index of \(R^ G\) in \(R\) coincides with a standard invariant for groups of automorphisms, the reduced order of \(G\).