Dobbs, David E.; Fontana, Marco Universally incomparable ring-homomorphisms. (English) Zbl 0535.13006 Bull. Aust. Math. Soc. 29, 289-302 (1984). 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. Cited in 1 ReviewCited in 11 Documents 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 Keywords:change of base; integral domain; universally incomparable homomorphisms; integral maps; radiciel maps PDFBibTeX XMLCite \textit{D. E. Dobbs} and \textit{M. Fontana}, Bull. Aust. Math. Soc. 29, 289--302 (1984; Zbl 0535.13006) Full Text: DOI References: [1] Kaplansky, Commutative rings (1974) [2] Grothendieck, Eléments de géométrie algébrique I (1971) [3] Gilmer, Multiplicative ideal theory (1972) · Zbl 0248.13001 [4] Gilmer, Pacific J. Math. 60 pp 81– (1975) · Zbl 0307.13011 · doi:10.2140/pjm.1975.60.81 [5] DOI: 10.1007/BF01796550 · Zbl 0443.13001 · doi:10.1007/BF01796550 [6] Dobbs, J. Algebra [7] Dobbs, Arch. Math. (Basel) [8] Demazure, Introduction to algebraic geometry and algebraic groups (1980) · Zbl 0431.14015 [9] Bourbaki, Commutative algebra (1972) [10] DOI: 10.2307/2033925 · Zbl 0145.27406 · doi:10.2307/2033925 [11] McAdam, Canad. J. Math. 23 pp 704– (1971) · Zbl 0223.13006 · doi:10.4153/CJM-1971-079-5 [12] DOI: 10.2307/2036800 · Zbl 0198.06001 · doi:10.2307/2036800 [13] Dobbs, Comment. Math. Univ. St. Pauli 31 pp 129– (1982) [14] Dobbs, Canad. Math. Bull. 23 pp 37– (1980) · Zbl 0432.13007 · doi:10.4153/CMB-1980-005-8 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.