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 RG\documentclass[12pt]{minimal}
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\begin{document}$$R^G$$\end{document} also has the property, where RG={r∈R|σ(r)=rfor allσ∈G},\documentclass[12pt]{minimal}
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\begin{document}$$R^G=\{r\in R \ | \ \sigma (r)=r \ \text {for all} \ \sigma \in G\},$$\end{document} the fixed ring of the action. In this paper we prove the following classes of rings are invariant under the operation R→RG:\documentclass[12pt]{minimal}
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\begin{document}$$R\rightarrow R^G:$$\end{document} locally pqr domains, Strong G-domains, G-domains, Hilbert rings, S-strong rings and root-closed domains. Further let P\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {P}$$\end{document} be a ring theoretic property and R⊆S\documentclass[12pt]{minimal}
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\begin{document}$$R\subseteq S$$\end{document} be a ring extension. A pair of rings (R, S) is said to be a P\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {P}$$\end{document}-pair, if T satisfies P\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {P}$$\end{document} for each intermediate ring R⊆T⊆S.\documentclass[12pt]{minimal}
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\begin{document}$$R\subseteq T\subseteq S.$$\end{document} We also prove that the property P\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {P}$$\end{document} descends from (R,S)→(RG,SG)\documentclass[12pt]{minimal}
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\begin{document}$$(R,S)\rightarrow (R^G, S^G)$$\end{document} in several cases. For instance, if P=\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {P}=$$\end{document} Going-down, Pseudo-valuation domain and “finite length of intermediate chains of domains”, we show each of these properties successfully transfer from (R,S)→(RG,SG).\documentclass[12pt]{minimal}
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\begin{document}$$(R,S)\rightarrow (R^G, S^G).$$\end{document}
