Constructing division rings as module-theoretic direct limits
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- by George M. Bergman PDF
- Trans. Amer. Math. Soc. 354 (2002), 2079-2114 Request permission
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.References
- J. L. Walsh, On interpolation by functions analytic and bounded in a given region, Trans. Amer. Math. Soc. 46 (1939), 46–65. MR 55, DOI 10.1090/S0002-9947-1939-0000055-0
- J. L. Walsh, On interpolation by functions analytic and bounded in a given region, Trans. Amer. Math. Soc. 46 (1939), 46–65. MR 55, DOI 10.1090/S0002-9947-1939-0000055-0
- George M. Bergman, Sfields finitely right-generated over subrings, Comm. Algebra 11 (1983), no. 17, 1893–1902. MR 709020, DOI 10.1080/00927878308822938
- P. M. Cohn, Universal algebra, 2nd ed., Mathematics and its Applications, vol. 6, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. MR 620952, DOI 10.1007/978-94-009-8399-1
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
- 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, DOI 10.1017/CBO9781139087193
- J. L. Fisher, Embedding free algebras in skew fields, Proc. Amer. Math. Soc. 30 (1971), 453–458. MR 281750, DOI 10.1090/S0002-9939-1971-0281750-9
- 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
- Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on torsion theories and dominant dimensions. MR 0284459, DOI 10.1007/BFb0061029
- Peter Malcolmson, A prime matrix ideal yields a skew field, J. London Math. Soc. (2) 18 (1978), no. 2, 221–233. MR 509937, DOI 10.1112/jlms/s2-18.2.221
- Peter Malcolmson, Determining homomorphisms to skew fields, J. Algebra 64 (1980), no. 2, 399–413. MR 579068, DOI 10.1016/0021-8693(80)90153-2
- A. H. Schofield, Representation of rings over skew fields, London Mathematical Society Lecture Note Series, vol. 92, Cambridge University Press, Cambridge, 1985. MR 800853, DOI 10.1017/CBO9780511661914
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- D. J. A. Welsh, Matroid theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0427112
- H. Whitney, On the abstract properties of linear dependence, Amer. J. Math. 57 (1935), 509–533.
Additional Information
- George M. Bergman
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
- Email: gbergman@math.berkeley.edu
- 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 - © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1881031