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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.
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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

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References

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