# Rings of invariants of finite groups when the bad primes exist

## TL;DR

This paper investigates the properties of rings of invariants under finite group actions, especially when bad primes exist, establishing conditions under which key ring properties are preserved or related.

## Contribution

It extends classical results on invariants of rings to cases with bad primes, showing how properties like radicals and semiprimeness relate between R and R^G.

## Key findings

- Jacobson radical of R^G equals intersection with R
- Semiprimeness of R implies semiprimeness of R^G
- Nilpotent trace in R implies R is nilpotent

## Abstract

Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not p-torsion free, is called the set of bad primes. When the ring is |G|-torsion free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are closely connected. The aim of the paper is to show that this is also true when B(R, G) is not empty set under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (resp., the prime radical) of the ring R^G is equal to the intersection of the Jacobson radical (resp., the prime radical) of R with R^G; if the ring R is semiprime then so is R^G; 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 R^G 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 R^G and \udim (R^G)\leq \udim (R)\leq |G| \udim (R^G).