E. Enochs has defined a ring A
(all rings are commutative, with 1) to be totally integrally closed (which we
abbreviate TIC) if for every integral extension h : B → C the induced map
h∗:Hom(C,A) →Hom(B,A) is surjective. Our main result here is that A is
TIC if and only if A is reduced, each residue class domain of A is normal
and has an algebraically closed field of fractions, and SpecA is extremal
(disjoint open sets have disjoint clopen neighborhoods). We use this fact to
settle negatively the open question, need a localization of a TIC ring be
TIC. The proofs depend on the following apparently new characterization
of extremal spaces: a topological space X is extremal if and only if there
is a Boolean algebra retraction of the family of all subsets of X onto the
family of all clopen subsets of X which takes every closed set into a subset of
itself.
