On characterizations of integrality involving the lying-over and incomparable properties
D. E. Dobbs · Journal of Commutative Algebra · 2009
The fact that residually algebraic pairs are the same as INC-pairs is generalized from the context of integral domains to that of arbitrary (commutative) rings.It is also shown that if A ⊆ B are rings with D the integral closure of A in B, then B is integral over A if and only if (A, B) is an INC-pair for which the extension D ⊆ B satisfies LO.However, a Noetherian local one-dimensional domain A is Henselian if and only if B is integral over A whenever B is a domain containing A such that (A, B) is an INC-pair for which the extension A ⊆ B satisfies LO. 1. Introduction.All rings considered in this note are commutative with identity, and all subrings are unital.Following [10, page 28] we let LO, INC and GU denote the lying-over, incomparable and goingup properties for ring extensions.If P is a property of (some) ring extensions and A ⊆ B are rings, we say that (A, B) is a P-pair in case D ⊆ E satisfies P for all rings A ⊆ D ⊆ E ⊆ B. The case of LO-pairs was introduced in [5], studied sporadically in the literature (cf.[12]), and recently given a new characterization in [3, Theorem 2.2].It was shown in [5, Corollary 3.2] that GU-pairs are the same as LO-pairs.As for INC-pairs, they were introduced and characterized (without the terminology) in [4, Corollary 4] and studied further, but only in the context of extensions of (commutative integral) domains, in [1].In particular, [1, Theorem 2.3] established that for extensions of domains, INC-pairs are the same as residually algebraic pairs.This domaintheoretic formulation has persisted in the summary of [1] given in the monograph [8], and several subsequent papers have also continued to study INC-pairs and residually algebraic pairs only for extensions of domains.Accordingly, our first order of business here is to generalize [1, Theorem 2.3] by showing that, for arbitrary ring extensions, the concepts of INC-pairs and residually algebraic pairs are equivalent.