Division rings and -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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a division ring with center and let denote the field of rational functions over . A square matrix is said to be totally transcendental over if the evaluation map , can be extended to . In this note it is shown that the tensor product is a -domain which has, up to isomorphism, a unique simple module iff any two totally transcendental matrices of the same order over 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.

**[1]**G. M. Bergman, Private communication, December 1981.**[2]**P. M. Cohn,*The similarity reduction of matrices over a skew field*, Math. Z.**132**(1973), 151-163. MR**0325646 (48:3993)****[3]**-,*Skew field constructions*, London Math. Soc. Lecture Notes Series, vol. 27, Cambridge Univ. Press, Cambridge, 1977. MR**0463237 (57:3190)****[4]**J. H. Cozzens,*Homological properties of the ring of differential polynomials*, Bull. Amer. Math. Soc.**76**(1970), 75-79. MR**0258886 (41:3531)****[5]**J. Cozzens and C. Faith,*Simple Noetherian rings*, Cambridge Univ. Press, Cambridge, 1975. MR**0396660 (53:522)****[6]**J. Hirschfeld and W. H. Wheeler,*Forcing, arithmetic, division rings*, Lecture Notes in Math., vol. 454, Springer-Verlag, Berlin and New York, 1975. MR**0389581 (52:10412)****[7]**N. Jacobson,*The structure of rings*, Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I., 1964. MR**0222106 (36:5158)****[8]**E. R. Kolchin,*Galois theory of differential fields*, Amer. J. Math.**75**(1953), 753-824. MR**0058591 (15:394a)****[9]**R. Resco, L. W. Small, and A. R. Wadsworth,*Tensor products of division rings and finite generation of subfields*, Proc. Amer. Math. Soc.**77**(1979), 7-10. MR**539619 (80g:16020)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
16A39,
16A33,
16A52

Retrieve articles in all journals with MSC: 16A39, 16A33, 16A52

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0875375-3

Keywords:
Division algebras,
-rings

Article copyright:
© Copyright 1987
American Mathematical Society