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

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