Totally integrally closed rings and extremal spaces

Abstract

E. Enochs has defined a ring A (all rings are commutative, with 1) to be totally integrally closed (which we abbreviate TIC) for every integral extension h: B—>C induced map /ι*: Horn (C, A) —»Horn (B, A) is surjective. Our main result here is that A is TIC and A is reduced, each residue class domain of A is normal and has an algebraically closed field of fractions, and Spec A is extremal (disjoint open sets have disjoint clopen neighborhoods). We use this fact to settle negatively open question, need a localization of a TIC ring be TIC. The proofs depend on following apparently new characterization of extremal spaces: a topological space X is extremal and there is a Boolean algebra retraction of family of all subsets of X onto family of all clopen subsets of X which takes every closed set into a subset of itself. 1* Basic facts* We summarize here some of facts about TIC rings from [2]. Most results are merely stated informally. The proofs are easy. However, we include simpler proofs of both Lemma 1 (which is, in substance, only if part of Theorem 1 of [2]) and of existence of the total integral closure of a reduced ring than are given in [2], and we also give some useful characterizations of TIC rings which are not stated (though they are implicit) in [2]. We use term normal for a domain integrally closed in its fraction field. When we speak of an integrally closed subring B of a ring C, we mean that B is integrally closed in C, rather than in, say, fraction field of B. Trivially, a product of TIC rings is TIC; a retract of a TIC ring is TIC; and an integrally closed subring of a TIC ring is TIC. Proposition 3 of [2] asserts that a domain is TIC and it is normal and has an algebraically closed fraction field. Hence, products of algebraically closed fields are TIC. Following [2], we define an extension h\A—*B to be tight if, equivalents, either (1) for each ideal IΦ (0) of B, h-^I) Φ (0); or (2) each nonzero element of B has a nonzero multiple in h(A); or (3) g: B —• C and g h(A) is injective then g is injective.