Rings of invariants of finite groups when the bad primes exist

Abstract

Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set of primes p such that and R is not p-torsion free, is called the set of bad primes. When the ring is -torsion free, i.e. the properties of the rings R and RG are closely connected. The aim of the article is to show that this is also true when under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (respectively, the prime radical) of the ring RG is equal to the intersection of the Jacobson radical (respectively, the prime radical) of R with RG; if the ring R is semiprime then so is RG; if the trace of the ring R is nilpotent then the ring itself is nilpotent; if R is a semiprime ring then R is left Goldie iff the ring RG is so, and in this case, the ring of G-invariants of the left quotient ring of R is isomorphic to the left quotient ring of RG and