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

Full-text PDF

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.

**1.**George M. Bergman,*Rational relations and rational identities in division algebras*, I, J. Algebra**43**(1976), 252-266. MR**55:56853a****2.**-,*Rational relations and rational identities in division algebras*, II, J. Algebra**43**(1976), 267-297. MR**55:56853b****3.**-,*Sfields finitely right-generated over subrings*, Comm. Algebra**11**(1983), 1893-1902. MR**85e:16032****4.**P. M. Cohn,*Universal Algebra*, 2nd ed., Reidel, Dordrecht, 1981. MR**82j:08001****5.**-,*Free Rings and their Relations*, 2nd edition, London Mathematical Society Monographs, No. 19, 1985. MR**87e:16006**. (First edition was London Mathematical Society Monograph No. 2, 1971; MR**51:8155**. Third edition is in preparation.)**6.**-,*Algebra*, 2nd ed. v. 3, Wiley & Sons, 1991. MR**92c:00001**(1st ed. MR**58:26625**).**7.**-,*Skew Fields. Theory of general division rings*, Encyclopedia of Mathematics and its Applications, v. 57, Cambridge Univ. Press. MR**97d:12003****8.**J. L. Fisher,*Embedding free algebras in skew fields*, Proc. Amer. Math. Soc.**30**(1971), 453-458. MR**43:7465****9.**Keith A. Kearnes,*Idempotent simple algebras*, Logic and Algebra (Pontignano, 1994), Lecture Notes in Pure and Appl. Math., v. 180, Dekker, 1996, pp. 529-572. MR**97k:08004****10.**Joachim Lambek,*Torsion Theories, Additive Semantics, and Rings of Quotients*, Springer Lecture Notes in Mathematics, v. 177, 1971. MR**44:1685****11.**Peter Malcolmson,*A prime matrix ideal yields a skew field*, J. London Math. Soc. (2)**18**(1978), 221-233. MR**80d:16003****12.**-,*Determining homomorphisms to skew fields*, J. Algebra**64**(1980), 399-413. MR**81k:16005****13.**A. H. Schofield,*Representation of Rings over Skew Fields*, London Mathematical Society Lecture Note Series, No. 92, 1985. MR**87c:16001****14.**B. L. van der Waerden,*Moderne Algebra*, Bd. I, Springer-Verlag, 1930. Jahrbuch**56**, p. 138. (There have been several later editions, and English translations, e.g.,*Modern Algebra*, v. I, transl. Fred Blum, Frederick Ungar Publishing Co., 1949, MR**10:587b**. Starting with the 1955 4th edition, the title was shortened to*Algebra*.)**15.**D. J. A. Welsh,*Matroid Theory*, London Mathematical Society Monographs, No. 8, 1976. MR**55:148****16.**H. Whitney,*On the abstract properties of linear dependence*, Amer. J. Math.**57**(1935), 509-533.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
05B35,
16K40,
16S90,
16E60,
16N80,
16S50,
16U20,
18A30

Retrieve articles in all journals with MSC (2000): 05B35, 16K40, 16S90, 16E60, 16N80, 16S50, 16U20, 18A30

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