Δ -Extension of rings and invariance properties of ring extension under group action

Abstract

Let R,T be commutative rings with identity such that R⊆T. A ring extension R⊆T is called a Δ-extension of rings if R1+R2 is a subring of T for each pair of subrings R1,R2 of T containing R. In this paper, a characterization of integrally closed Δ-extension of rings is given. The equivalence of Δ-extension of rings and λ-extension of rings is established for an integrally closed extension of a local ring. Over a finite dimensional, integrally closed extension of local rings, the equivalence of Δ-extensions of rings, FIP, and FCP is shown. Let R be a subring of T such that R is invariant under action by G, where G is a subgroup of the automorphism group of T. If R⊆T is a Δ-extension of rings, then RG⊆TG is a Δ-extension of rings under some conditions. Many such G-invariant properties are also discussed.