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On chain morphisms of commutative rings

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Résumé

On dit qu’un homomorphismef :A →B d’anneaux commutatifs est un morphisme de chaîne (resp., den-chaîne pour un entiern ≥ 1) si toute chaîne d’idéaux premiers (resp., d’au plusn idéaux premiers) deA se relève en une chaîne d’idéaux premiers deB. Sif est un morphisme den-chaîne, alorsf n’est pas forcément un morphisme de (n + 1)-chaîne, même siA etB sont des anneaux intègres, doncf n’est pas un morphisme de chaîne. Sif est un morphisme den-chaîne pour toutn, alorsf est un morphisme de chaîne. Un morphisme de chaîne n’est pas forcément un morphisme de chaîne universel. Pour tout entiern ≥ 2,f est universellement un morphisme den-chaîne si et seulement sif est universellement un morphisme de chaîne. Un morphisme qui est universellement de chaîne et universellement incomparable n’est pas nécessairement entier, même siA etB sont des anneaux intègres de dimension 1 (au sens de Krull).

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, 37996-1300, Knoxville, Tennessee, U.S.A.

    David E. Dobbs & Andrew J. Hetzel

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  1. David E. Dobbs
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  2. Andrew J. Hetzel
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Dobbs, D.E., Hetzel, A.J. On chain morphisms of commutative rings. Rend. Circ. Mat. Palermo 53, 71–84 (2004). https://doi.org/10.1007/BF02921428

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