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Universally Incomparable Ring-Homomorphisms

Published online by Cambridge University Press:  17 April 2009

David E. Dobbs  and
Marco Fontana
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996, USA;
Marco Fontana
Affiliation:
Dipartimento di Matematica, Università di Roma I, 001 85 Roma, Italy.
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Abstract

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A homomorphism f: R → T of (commutative) rings is said to be universally incomparable in case each base change R → S induces an incomparable map S → S⊗RT. The most natural examples of universally incomparable homomorphisms are the integral maps and radiciel maps. It is proved that a homomorphism f: R → T is universally incomparable if and only if f is an incomparable map which induces algebraic field extensions of fibres, k(f-1(Q))→k(Q), for each prime ideal Q of T. In several cases (f algebra-finite, T generated as R-algebra by primitive elements, T an overring of a one-dimensional Noetherian domain R), each universally incomparable map is shown to factor as a composite of an integral map and a special kind of radiciel.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 3 , June 1984 , pp. 289 - 302
Copyright
Copyright © Australian Mathematical Society 1984

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