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 \((R,S)\rightarrow (R^G, S^G).\)