On λ-extensions of commutative rings

Abstract

Let [Formula: see text] be commutative rings with identity such that [Formula: see text]. We recall that [Formula: see text] is called a [Formula: see text]-extension of rings if the set of all subrings of [Formula: see text] containing [Formula: see text] (the “intermediate rings”) is linearly ordered under inclusion. In this paper, a characterization of integrally closed [Formula: see text]-extension of rings is given. For example, we show that if [Formula: see text] is a local ring, then [Formula: see text] is an integrally closed [Formula: see text]-extension of rings if and only if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] is a valuation domain. 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.