Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T19:37:33.159Z Has data issue: false hasContentIssue false
  • English
  • Français

On a construction of the universal field of fractions of a free algebra

Published online by Cambridge University Press:  26 February 2010

G. Revesz
G. Revesz
Affiliation:
Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045, U.S.A.
Get access

Extract

In this note we present a simple way of obtaining the universal field of fractions of certain free rings as a subfield of an ultrapower of a (by no means unique) skew field. This method of embedding was discovered by Amitsur in [1]; our presentation uses Cohn's specialization lemma and the embedding is constructed in terms of full matrices over the rings in question (Theorem 3.2). In particular, if k is an infinite commutative field, the universal field of fractions of a free k-algebra can be realized as a subfield of an ultrapower of any skew extension of k, with centre k, which is infinite dimensional over k. Thus many problems concerning the universal field of fractions of a free k-algebra can be settled by studying skew extensions of k of relatively simple structure. More precisely and more generally, let E be a skew field with centre k and denote by R the free E-ring on X over k. Write U for the universal field of fractions of R and assume that U embeds in an ultrapower of a skew field D. Then by Łos' theorem, U inherits the first-order properties of D which can be expressed by universal sentences.

Type
Research Article
Information
Mathematika , Volume 31 , Issue 2 , December 1984 , pp. 227 - 233
Copyright
Copyright © University College London 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Amitsur, S. A.. Rational identities and applications to algebra and geometry. J. Algebra, 3 (1966), 304–359.CrossRefGoogle Scholar
2.Cohn, P. M.. Free rings and their relations (Academic Press, London, New York, 1971).Google Scholar
3.Cohn, P. M.. Skew field constructions (Cambridge University Press, 1977).Google Scholar
4.Cohn, P. M.. Centralisateurs dans les corps libres, École de Printemps d'informatique Theorique “Series formelles en variables non commutatives et applications”, Vieux-Boucau les Bains (Landes), 23–27 mai 1977.Google Scholar
5.Cohn, P. M.. The universal field of fractions of a semifir III. Centralizers and normalizers. Proc. London Math. Soc. (3), 50 (1985), 95–113.CrossRefGoogle Scholar
6.Cohn, P. M.. On a theorem from “Skew field constructions”. Proc. Amer. Math. Soc, 84 (1982), 1–7.Google Scholar