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
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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.References
- G. M. Bergman, Private communication, December 1981.
- Paul M. Cohn, The similarity reduction of matrices over a skew field, Math. Z. 132 (1973), 151–163. MR 325646, DOI 10.1007/BF01213920
- Paul Moritz Cohn, Skew field constructions, London Mathematical Society Lecture Note Series, No. 27, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0463237
- John H. Cozzens, Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79. MR 258886, DOI 10.1090/S0002-9904-1970-12370-9
- John Cozzens and Carl Faith, Simple Noetherian rings, Cambridge Tracts in Mathematics, No. 69, Cambridge University Press, Cambridge-New York-Melbourne, 1975. MR 0396660, DOI 10.1017/CBO9780511565700
- Joram Hirschfeld and William H. Wheeler, Forcing, arithmetic, division rings, Lecture Notes in Mathematics, Vol. 454, Springer-Verlag, Berlin-New York, 1975. MR 0389581, DOI 10.1007/BFb0064082
- Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
- E. R. Kolchin, Galois theory of differential fields, Amer. J. Math. 75 (1953), 753–824. MR 58591, DOI 10.2307/2372550
- Richard Resco, Lance W. Small, and Adrian R. Wadsworth, Tensor products of division rings and finite generation of subfields, Proc. Amer. Math. Soc. 77 (1979), no. 1, 7–10. MR 539619, DOI 10.1090/S0002-9939-1979-0539619-5
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