Density theorems and its applications
https://doi.org/10.26907/2949-3919.2023.4.3-34
Abstract
This article is semi-review and is also methodological in nature. The article discusses generalizations of the Jacobson-Chevalley density theorem, on the basis of which a number of results from linear algebra are presented, related to centralizers of locally algebraic linear operators. Also, based on Jacobson’s approach to constructing Galois theory based on the density theorem, a proof of Hilbert’s theorem 90 and some of its well-known generalizations are given.
Supporting agencies: The work is supported by the Russian Science Foundation (grant no. 22-21-00267) and performed under the development program of Volga Region Mathematical Center (agreement no. 075-02-2023-944).
About the Author
A. N. Abyzov
Kazan Federal University
Russian Federation
Russian Federation
Adel Nailevich Abyzov
18 Kremlyovskaya str., Kazan 420008, Russia
References
1. T. Nakayma, Uber einfache distributive Systeme unendlicher R¨ange ¨ , Proc. Imp. Acad.Tokyo 20, 61–66 (1944).
2. N. Jacobson, Structure theory of simple rings without finiteness assumptions, Trans. Amer. Math. Soc., 57 (2), 228–245 (1945). DOI: https://doi.org/10.2307/1990204
3. N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ. 37, AMS, Providence R.I., 1956.
4. N. Jacobson, Basic algebra II: Second edition, W.H. Freeman and Company, New York, 1989.
5. N. Bourbaki, Elements of the history of mathematics, Springer-Verlag, Berlin, 1994.
6. E. Artin, Galois theory, Notre Dame Math. Lectures 2, 1971.
7. K. Fuller, Density and equivalence, J. Algebra 29 (3), 528–550 (1974). DOI: https://doi.org/10.1016/0021-8693(74)90088-X
8. B.J. M`uller, On Morita duality, Canad. J. Math. 21, 1338–1347 (1969). DOI: https://doi.org/10.4153/CJM-1969-147-7
9. P.A. Krylov, A.A. Tuganbaev, Modules over discrete valuation domains, De Gruyter, Berlin, New York, 2008. DOI: https://doi.org/10.1515/9783110205787
10. C. Faith, Algebra: rings, modules and categories I, Springer-Verlag, Berlin, Heidelberg, 1973. DOI: https://doi.org/10.1007/978-3-642-80634-6
11. M.C. Iovanov, Z. Mesyan, M.L. Reyes, Infinite-Dimensional Diagonalization and Semisimplicity, Israel J. Math. 215, 801–855 (2016). DOI: https://doi.org/10.1007/s11856-016-1395-5
12. N. Jacobson, A note on division rings, Am. J. Math. 69 (1), 27–36 (1947). DOI: https://doi.org/10.2307/2371651
13. H. Cartan, Les principaux th´eor`emes de la th´eorie de Galois pour les corps non n´ecessairement commutatifs, Comptes rendus de l’Acad´emie des Sciences 224, 249–251 (1947).
14. S.Warner, Topological rings, North Holland, 1993.
15. Kaplansky, Infinite abelian groups, Univ. of Michigan Press, Ann Arbor, MI, 1954.
16. H. Furstenberg, On the infinitude of primes, Am. Math. Mon. 62 (5), 353 (1957). DOI: https://doi.org/10.2307/2307043
17. Е.В. Новоселов, Введение в полиадический анализ, ПГУ, Петрозаводск, 1982.
18. Z. Mesyan, Infinite-dimensional triangularization, J. Pure Appl. Algebra 222 (7), 1529– 1547 (2018). DOI: https://doi.org/10.1016/j.jpaa.2017.07.010
19. G. Dolinar, A.E. Guterman, B. Kuzma, P. Oblak, Extremal matrix centralizers, Linear Algebra Appl. 438 (7), 2904–2910 (2013). DOI: https://doi.org/10.1016/j.laa.2012.12.010
20. Abyzov, A. Maklakov, Locally algebraic linear operators and their centralizers, Linear Algebra Appl. 662, 1–17 (2023). DOI: https://doi.org/10.1016/j.laa.2022.12.022
21. P. Semrl, ˇ Non-linear commutativity preserving maps, Acta Sci. Math. (Szeged) 71 (3–4), 781–819 (2005).
22. P. Semrl, ˇ Maps on matrix spaces, Linear Algebra Appl. 413 (2–3), 364–393 (2006). DOI: https://doi.org/10.1016/j.laa.2005.03.011
23. Abyzov, A. Maklakov, Automorphisms of dense subrings of the endomorphism ring of a free module, Linear Algebra Appl. 679, 220–230 (2023). DOI: https://doi.org/10.1016/j.laa.2023.09.021
24. J. Courtemanche, M. Dugas, Automorphisms of the endomorphism algebra of a free module, Linear Algebra Appl. 510, 79–91 (2016). DOI: https://doi.org/10.1016/j.laa.2016.08.017
25. I.G. Connell, Elementary generalizations of Hilbert’s Theorem 90, Canad. Math. Bull. 8 (6), 749–757 (1965). DOI: https://doi.org/10.4153/CMB-1965-055-2
26. Speiser, Zahlentheoretische S¨atze aus der Gruppentheorie, Math. Zeit. 5, 1–6 (1919).
27. E. Noether, Der Hauptgeschlechtssatz f¨ur relativ-galoissche Zahlk¨orper, Math. Ann. 108, 411–419 (1933).
Review
For citations:
Abyzov A.N. Density theorems and its applications. Mathematics and Theoretical Computer Science. 2023;1(4):3-34. (In Russ.) https://doi.org/10.26907/2949-3919.2023.4.3-34
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