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A dimension theorem for division rings

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Abstract

LetD be a division algebra over a fieldk, letn be an arbitrary positive integer, and letk(x 1,...,x n) denote the rational function field inn variables overk. In this note we complete previous work by proving that the following three conditions are equivalent: (i) there exists an integerj such that the matrix ringM j(D) contains a commutative subfield which has transcendence degreen overk; (ii) K dim (D⊗k k(x 1,...,x n )) =n; (iii) gl. dim (D⊗k k(x 1,...,x n )) =n. The crucial tool in the proof of this theorem is the Nullstellensatz forD[x 1,...,x n] which was obtained by Amitsur and Small.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, 90007, Los Angeles, California, USA

    Richard Resco

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  1. Richard Resco
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Resco, R. A dimension theorem for division rings. Israel J. Math. 35, 215–221 (1980). https://doi.org/10.1007/BF02761192

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  • DOI: https://doi.org/10.1007/BF02761192

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