Descent of properties of rings and pairs of rings to fixed rings

Abstract

Let $G$ be a group acting via ring automorphisms on an integral domain $R.$ A ring-theoretic property of $R$ is said to be $G$-invariant, if $R^G$ also has the property, where $R^G=\{r\in R \ | \ \sigma(r)=r \ \text{for all} \ \sigma\in G\},$ the fixed ring of the action. In this paper we prove the following classes of rings are invariant under the operation $R\rightarrow R^G:$ locally pqr domains, Strong G-domains, G-domains, Hilbert rings, $S$-strong rings and root-closed domains. Further let $\mathscr{P}$ be a ring theoretic property and $R\subseteq S$ be a ring extension. A pair of rings $(R,S)$ is said to be a $\mathscr{P}$-pair, if $T$ satisfies $\mathscr{P}$ for each intermediate ring $R\subseteq T\subseteq S.$ We also prove that the property $\mathscr{P}$ descends from $(R,S)\rightarrow (R^G, S^G)$ in several cases. For instance, if $\mathscr{P}=$ Going-down, Pseudo-valuation domain and "finite length of intermediate chains of domains", we show each of these properties successfully transfer from $(R,S)\rightarrow (R^G, S^G).$