Universally Incomparable Ring-Homomorphisms

Abstract

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.