Vol. 32, No. 3, 1970

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Totally integrally closed rings and extremal spaces

Melvin Hochster

Vol. 32 (1970), No. 3, 767–779

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.

Mathematical Subject Classification
Primary: 13.80
Received: 10 April 1969
Published: 1 March 1970
Melvin Hochster
Department of Mathematics
University of Michigan
East Hall, 530 Church Street
Ann Arbor MI 48109-1043
United States