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

DOI:
https://doi.org/10.1090/S0002-9947-02-02927-6

Published electronically:
January 8, 2002

MathSciNet review:
1881031

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Abstract | References | Similar Articles | Additional Information

Abstract: If is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring generated by a homomorphic image of is a rule putting on all free -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 may be constructed from this data. (A classical precursor of this is the construction of as a field with additive group a direct limit of copies of .)

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:
https://doi.org/10.1090/S0002-9947-02-02927-6

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