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

TL;DR

This paper investigates how certain ring properties are preserved under fixed ring operations and ring extensions with group actions, establishing invariance and descent results for various classes of rings and properties.

Contribution

It proves that multiple classes of rings are invariant under fixed ring formation and that certain properties descend through fixed ring extensions under group actions.

Findings

Locally pqr domains, G-domains, and Hilbert rings are invariant under fixed ring operation.

Properties like Going-down and Pseudo-valuation domains descend from rings to their fixed rings.

Several properties transfer successfully from ring extensions to fixed rings under group actions.

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