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Universally incomparable ring-homomorphisms. (English) Zbl 0535.13006

A homomorphism \(f:R\to T\) of (commutative) rings is said to be universally incomparable in case each base change \(R\to S\) induces an incomparable map \(S\to S\otimes_ RT.\) The most natural examples of universally incomparable homomorphisms are the integral maps and radiciel maps. It is proved that a homomorphism \(f:R\to T\) is universally incomparable if and only if f is an incomparable map which induces algebraic field extensions of fibres, \(k(f^{-1}(Q))\to k(Q)\), 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.

MSC:

13B02 Extension theory of commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13A15 Ideals and multiplicative ideal theory in commutative rings
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