Constructing division rings as moduletheoretic direct limits
Author:
George M. Bergman
Journal:
Trans. Amer. Math. Soc. 354 (2002), 20792114
MSC (2000):
Primary 05B35, 16K40, 16S90; Secondary 16E60, 16N80, 16S50, 16U20, 18A30
Published electronically:
January 8, 2002
MathSciNet review:
1881031
Fulltext PDF Free Access
Abstract 
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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), 252266. MR 55:56853a
 2.
, Rational relations and rational identities in division algebras, II, J. Algebra 43 (1976), 267297. MR 55:56853b
 3.
George
M. Bergman, Sfields finitely rightgenerated over subrings,
Comm. Algebra 11 (1983), no. 17, 1893–1902. MR 709020
(85e:16032), http://dx.doi.org/10.1080/00927878308822938
 4.
P.
M. Cohn, Universal algebra, 2nd ed., Mathematics and its
Applications, vol. 6, D. Reidel Publishing Co., DordrechtBoston,
Mass., 1981. MR
620952 (82j:08001)
 5.
P.
M. Cohn, Free rings and their relations, 2nd ed., London
Mathematical Society Monographs, vol. 19, Academic Press, Inc.
[Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
(87e:16006)
P.
M. Cohn, Free rings and their relations, Academic Press,
LondonNew York, 1971. London Mathematical Society Monographs, No. 2. MR 0371938
(51 #8155)
 6.
P.
M. Cohn, Algebra. Vol. 3, 2nd ed., John Wiley & Sons,
Ltd., Chichester, 1991. MR 1098018
(92c:00001)
Paul
Moritz Cohn, Algebra. Vol. 2, John Wiley & Sons,
LondonNew YorkSydney, 1977. With errata to Vol. I. MR 0530404
(58 #26625)
 7.
P.
M. Cohn, Skew fields, Encyclopedia of Mathematics and its
Applications, vol. 57, Cambridge University Press, Cambridge, 1995.
Theory of general division rings. MR 1349108
(97d:12003)
 8.
J.
L. Fisher, Embedding free algebras in skew
fields, Proc. Amer. Math. Soc. 30 (1971), 453–458. MR 0281750
(43 #7465), http://dx.doi.org/10.1090/S00029939197102817509
 9.
Keith
A. Kearnes, Idempotent simple algebras, Logic and algebra
(Pontignano, 1994) Lecture Notes in Pure and Appl. Math., vol. 180,
Dekker, New York, 1996, pp. 529–572. MR 1404955
(97k:08004)
 10.
Joachim
Lambek, Torsion theories, additive semantics, and rings of
quotients, With an appendix by H. H. Storrer on torsion theories and
dominant dimensions. Lecture Notes in Mathematics, Vol. 177,
SpringerVerlag, BerlinNew York, 1971. MR 0284459
(44 #1685)
 11.
Peter
Malcolmson, A prime matrix ideal yields a skew field, J.
London Math. Soc. (2) 18 (1978), no. 2,
221–233. MR
509937 (80d:16003), http://dx.doi.org/10.1112/jlms/s218.2.221
 12.
Peter
Malcolmson, Determining homomorphisms to skew fields, J.
Algebra 64 (1980), no. 2, 399–413. MR 579068
(81k:16005), http://dx.doi.org/10.1016/00218693(80)901532
 13.
A.
H. Schofield, Representation of rings over skew fields, London
Mathematical Society Lecture Note Series, vol. 92, Cambridge
University Press, Cambridge, 1985. MR 800853
(87c:16001)
 14.
B.
L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar
Publishing Co., New York, N. Y., 1949. Translated from the second revised
German edition by Fred Blum; With revisions and additions by the author. MR 0029363
(10,587b)
 15.
D.
J. A. Welsh, Matroid theory, Academic Press [Harcourt Brace
Jovanovich, Publishers], LondonNew York, 1976. L. M. S. Monographs, No. 8.
MR
0427112 (55 #148)
 16.
H. Whitney, On the abstract properties of linear dependence, Amer. J. Math. 57 (1935), 509533.
 1.
 George M. Bergman, Rational relations and rational identities in division algebras, I, J. Algebra 43 (1976), 252266. MR 55:56853a
 2.
 , Rational relations and rational identities in division algebras, II, J. Algebra 43 (1976), 267297. MR 55:56853b
 3.
 , Sfields finitely rightgenerated over subrings, Comm. Algebra 11 (1983), 18931902. 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), 453458. 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. 529572. 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), 221233. MR 80d:16003
 12.
 , Determining homomorphisms to skew fields, J. Algebra 64 (1980), 399413. 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, SpringerVerlag, 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), 509533.
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Additional Information
George M. Bergman
Affiliation:
Department of Mathematics, University of California, Berkeley, California 947203840
Email:
gbergman@math.berkeley.edu
DOI:
http://dx.doi.org/10.1090/S0002994702029276
PII:
S 00029947(02)029276
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 23year 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
