Abstract

Let E*G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E*G-isomorphism, there exists at most one Hughes-free division E*G-ring. However, the existence of a Hughes-free division E*G-ring {mathcal {D}}_{E*G} for an arbitrary locally indicable group G is still an open question. Nevertheless, {mathcal {D}}_{E*G} exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether {mathcal {D}}_{E*G} is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists {mathcal {D}}_{E[G]} and it is universal. In Appendix we give a description of {mathcal {D}}_{E[G]} when G is a RFRS group.

Highlights

  • A division R-ring φ : R → D is called epic if φ(R) generates D as a division ring

  • Given a ring R, Cohn introduced the notion of universal division R-ring

  • The universal division R-ring D is called universal division ring of fractions of R if D is epic and rkD is faithful

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Summary

Introduction

A division R-ring φ : R → D is called epic if φ(R) generates D as a division ring. Each division R-ring D induces a Sylvester matrix rank function rkD on R. Given a ring R, Cohn introduced the notion of universal division R-ring (see, for example, [4, Section 7.2]). In the language of Sylvester rank functions, an epic division R-ring D is universal if for every division R-ring E, rkD ≥ rkE. By a result of Cohn [3, Theorem 4.4.1], the universal epic division R-ring is unique up to R-isomorphism. The universal division R-ring D is called universal division ring of fractions of R if D is epic and rkD is faithful (that is R is embedded in D)

Jaikin-Zapirain
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Notation and definitions
Ordered groups
Hughes-free division rings
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Sylvester matrix rank functions
Von Neumann rank function
The natural extension
A general criterion of universality
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A criterion for a group to be Lewin
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Examples of Lewin groups
Universality of rkG
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Characters
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Novikov rings
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Invertibility over Novikov rings
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Full Text
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