Properties of rings and of ring extensions that are invariant under group action

Abstract

We expand the work in invariant theory inspired by Hilbert's Fourteenth Problem. Given a commutative ring with identity $R$ and a subgroup $G$ of the automorphism group of $R$, the \textit{fixed ring} is $R^G:=\{r\in R\,|\,\sigma(r)=r\;\text{for all}\;\sigma\in G\}$. That is, $R^G$ is the collection of elements of $R$ that are fixed by all automorphisms in $G$. Properties of $R$ inherited by $R^G$ and properties of the extension $R^G\subseteq R$ have been studied extensively. We call properties of $R$ that are inherited by $R^G$ \textit{invariant (under the group action by $G$)}.