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

Abstract

Let [Formula: see text] be commutative rings with identity such that [Formula: see text]. A ring extension [Formula: see text] is called a [Formula: see text]-extension of rings if [Formula: see text] is a subring of [Formula: see text] for each pair of subrings [Formula: see text] of [Formula: see text] containing [Formula: see text]. In this paper, a characterization of integrally closed [Formula: see text]-extension of rings is given. The equivalence of [Formula: see text]-extension of rings and [Formula: see text]-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 [Formula: see text]-extensions of rings, FIP, and FCP is shown. Let [Formula: see text] be a subring of [Formula: see text] such that [Formula: see text] is invariant under action by [Formula: see text], where [Formula: see text] is a subgroup of the automorphism group of [Formula: see text]. If [Formula: see text] is a [Formula: see text]-extension of rings, then [Formula: see text] is a [Formula: see text]-extension of rings under some conditions. Many such [Formula: see text]-invariant properties are also discussed.