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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Constructing division rings as module-theoretic direct limits


Author: George M. Bergman
Journal: Trans. Amer. Math. Soc. 354 (2002), 2079-2114
MSC (2000): Primary 05B35, 16K40, 16S90; Secondary 16E60, 16N80, 16S50, 16U20, 18A30
Published electronically: January 8, 2002
MathSciNet review: 1881031
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Abstract: If $R$ is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring $D$ generated by a homomorphic image of $R$ is a rule putting on all free $R$-modules of finite rank matroid structures (closure operators satisfying the exchange axiom) subject to certain functoriality conditions. This note gives a new description of how $D$ may be constructed from this data. (A classical precursor of this is the construction of $\mathbf Q$ as a field with additive group a direct limit of copies of $\mathbf Z$.)

The division rings of fractions of right and left Ore rings, the universal division ring of a free ideal ring, and the concept of a specialization of division rings are then interpreted in terms of this construction.


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Additional Information

George M. Bergman
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: gbergman@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02927-6
PII: S 0002-9947(02)02927-6
Received by editor(s): January 24, 2000
Received by editor(s) in revised form: August 23, 2001
Published electronically: January 8, 2002
Additional Notes: Part of this work was done in 1977 while the author was supported by the Miller Institute for Basic Research in the Sciences
The author apologizes to workers in the field who were inconvenienced by the 23-year delay between his getting the main result of this paper, and his finding the time to prepare it for publication
Article copyright: © Copyright 2002 American Mathematical Society