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Proceedings of the American Mathematical Society

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

 

 

Division rings and $ V$-domains


Author: Richard Resco
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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0875375-3
Keywords: Division algebras, $ V$-rings
Article copyright: © Copyright 1987 American Mathematical Society