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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Division rings and $V$-domains
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by Richard Resco
Proc. Amer. Math. Soc. 99 (1987), 427-431
DOI: https://doi.org/10.1090/S0002-9939-1987-0875375-3

Abstract:

Let $D$ be a division ring with center $k$ and let $k\left ( x \right )$ denote the field of rational functions over $k$. A square matrix $\tau \in {M_n}\left ( D \right )$ is said to be totally transcendental over $k$ if the evaluation map $\varepsilon : k\left [ x \right ] \to {M_n}\left ( D \right ),\varepsilon \left ( f \right ) = f\left ( \tau \right )$, can be extended to $k\left ( x \right )$. In this note it is shown that the tensor product $D{ \otimes _k}k\left ( x \right )$ is a $V$-domain which has, up to isomorphism, a unique simple module iff any two totally transcendental matrices of the same order over $D$ are similar. The result applies to the class of existentially closed division algebras and gives a partial solution to a problem posed by Cozzens and Faith.
References
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Bibliographic Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 99 (1987), 427-431
  • MSC: Primary 16A39; Secondary 16A33, 16A52
  • DOI: https://doi.org/10.1090/S0002-9939-1987-0875375-3
  • MathSciNet review: 875375