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

摘要

Let R, T be commutative rings with identity such that R subset of T. A ring extension R subset of T is called a Delta-extension of rings if R-1 + R-2 is a subring of T for each pair of subrings R-1, R-2 of T containing R. In this paper, a characterization of integrally closed Delta-extension of rings is given. The equivalence of Delta-extension of rings and lambda-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 Delta-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 subset of T is a Delta-extension of rings, then R-G subset of T-G is a Delta-extension of rings under some conditions. Many such G-invariant properties are also discussed.