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Commutative rings and binomial coefficients

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Abstract

We characterize commutative domainsR for which theR-module ofR-valued polynomials is generated by binomial coefficients. This turns out to be a special case of a more general result concerning commutative ringsR of zero characteristics in which fork=1,2,... and allx∈R the productx(x−1)·­.·(x−k+1) is divisible byk! inR.

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References

  1. Gerboud, G.: Exemples d'anneauxA pour lesquels\(((\begin{array}{*{20}c} X \\ n \\ \end{array} ))_{n \in N} \) est une base duA-module des polynomes entiéres surA. Comptes Rendus Acad. Sci. Paris307, 1–4 (1988).

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Authors and Affiliations

  1. Institut für Mathematik, Karl-Franzens-Universität, Heinrichstrasse 36/IV, A-8010, Graz, Austria

    F. Halter-Koch

  2. Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, PL-50-384, Wrocław, Poland

    W. Narkiewicz

Authors
  1. F. Halter-Koch
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  2. W. Narkiewicz
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The work of the second author has been sponsored by the KBN grant 2 1037 91 01

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Halter-Koch, F., Narkiewicz, W. Commutative rings and binomial coefficients. Monatshefte für Mathematik 114, 107–110 (1992). https://doi.org/10.1007/BF01535576

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  • DOI: https://doi.org/10.1007/BF01535576

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