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 $\mathcal{B}(R,G)$ 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 $\left|G\right|$-torsion free, i.e. $\mathcal{B}(R,G)=\varnothing ,$ the properties of the rings R and R^G are closely connected. The aim of the article is to show that this is also true when $\mathcal{B}(R,G)\ne \varnothing$ under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (respectively, the prime radical) of the ring R^G is equal to the intersection of the Jacobson radical (respectively, 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 $\text{udim}(R^G)\le \text{udim}(R)\le \left|G\right|\text{udim}(R^G).$

Keywords:

• Bad primes
• group of automorphisms
• nilpotent ideal
• semiprime ring
• semisimple Artinian ring
• the Jacobson radical
• the prime radical
• the ring of invariants

Mathematics Subject Classification 2010:

• 16W22
• 20C05
• 16N20
• 16N60

Acknowledgments

This work was done during the visit of the first author to the University of São Paulo whose hospitality and support are greatly acknowledged.

Additional information

Funding

VB is partly supported by Fapesp grant (2017/02946-0). VF is partly supported by CNPq grant (304467/2017-0) and by Fapesp grant (2014/09310-5).