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Fragmented domains have infinite Krull dimension

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Résumé

On dit qu'un anneau intègreR est fragmenté si pour tout élément non-inversibler deR, il existe un élément non-inversibles deR tel que r∈∩Rs n. On montre, pour un anneau intègreR qui n'est pas un corps, qu'il existe un idéal maximal deR qui contient une chaîne strictement croissante d'idéaux premiers deR. Si, de plus,R n'ax qu'un nombre fini d'idéaux maximaux, alors on peut reformuler l'affirmation précédente pour tout idéal maximal deR. Il découle que toute anneau intègreR, qui n'est pas un corps et qui possède un idéal premierP tel queR+PR p soit fragmenté, doit être de dimension infinie (au sens de Krull). On donne un exemple d'un tel anneauR qui n'est pas fragmenté.

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Authors and Affiliations

  1. Department of Mathematics, North Dakota State University, 58105-5075, Fargo, ND, USA

    Jim Coykendall

  2. Department of Mathematics, University of Tennessee, 37996-1300, Knoxville, TN, USA

    David Dobbs

Authors
  1. Jim Coykendall
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  2. David Dobbs
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Coykendall, J., Dobbs, D. Fragmented domains have infinite Krull dimension. Rend. Circ. Mat. Palermo 50, 377–388 (2001). https://doi.org/10.1007/BF02844993

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