Abstract
Using finite topologies defined on the algebra of linear operators, we investigate centralizers and double centralizers of locally algebraic linear operators. In particular, for an arbitrary locally algebraic operator \(A\), we establish the conditions under which the equality \(CC(A) = C(A)\) is fulfilled and, in the case of an algebraically closed field, we describe minimal locally algebraic linear operators. We have studied automorphisms of dense in finite topology subrings of the rings of endomorphisms of free modules over projectively free rings.
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REFERENCES
M. C. Iovanov, Z. Mesyan, and M. L. Reyes, “Infinite-dimensional diagonalization and semisimplicity,” Isr. J. Math. 215, 801–855 (2016). https://doi.org/10.1007/s11856-016-1395-5
Z. Mesyan, “Infinite-dimensional triangularization,” J. Pure Appl. Algebra 222, 1529–1547 (2018). https://doi.org/10.1016/j.jpaa.2017.07.010
N. Jacobson, Lectures in Abstract Algebra: II. Linear Algebra, Graduate Texts in Mathematics, Vol. 31 (Springer, New York, 1975). https://doi.org/10.1007/978-1-4684-7053-6
N. Jacobson, Structure of Rings, American Mathematical Society Colloquim Publications, Vol. 37 (American Mathematical Society, Providence, R.I., 1956).
O. Bezushchak, “Automorphisms and derivations of algebras of infinite matrices,” Linear Algebra Its Appl. 650, 42–59 (2022). https://doi.org/10.1016/j.laa.2022.05.020
J. Courtemanche and M. Dugas, “Automorphisms of the endomorphism algebra of a free module,” Linear Algebra Its Appl. 510, 79–91 (2016). https://doi.org/10.1016/j.laa.2016.08.017
I. M. Isaacs, “Automorphisms of matrix algebras over commutative rings,” Linear Algebra Its Appl. 31, 215–231 (1980). https://doi.org/10.1016/0024-3795(80)90221-9
P. A. Krylov and A. A. Tuganbaev, Modules over Discrete Valuation Domains, De Gruyter Expositions in Mathematics, Vol. 43 (De Gruyter, Berlin, 2018). https://doi.org/10.1515/9783110205787
S. Warner, Topological Rings, North-Holland Mathematical Studies, Vol. 178 (North-Holland, Amsterdam, 1993).
A. Kostić, Z. Z. Petrović, Z. S. Pucanović, and M. Roslavcev, “On a generalized Jordan form of an infinite upper triangular matrix,” Linear Multilinear Algebra 69, 1534–1542 (2021). https://doi.org/10.1080/03081087.2019.1632783
R. Słowik, “Corrigendum to ‘Every infinite triangular matrix is similar to a generalized infinite Jordan matrix’ (Linear Multilinear Algebra 65 (2017), 1362–1373),” Linear Multilinear Algebra 66, 1278 (2018). https://doi.org/10.1080/03081087.2017.1397094
G. Dolinar, A. E. Guterman, B. Kuzma, and P. Oblak, “Extremal matrix centralizers,” Linear Algebra Its Appl. 438, 2904–2910 (2013). https://doi.org/10.1016/j.laa.2012.12.010
P. Šemrl, “Non-linear commutativity preserving maps,” Acta Sci. Math. (Szeged) 71, 781–819 (2005).
H. Furstenberg, “On the infinitude of primes,” Am. Math. Monthly 62, 353 (1955).
P. M. Cohn, Free Rings and Their Relations, L. M. S. Monographs (Academic, London, 1971).
I. Kaplansky, Commutative Rings (The Univ. of Chicago Press, Chicago, 1970).
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Translated by E. Seifina
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Abyzov, A.N., Maklakov, A.D. Finite Topologies and Their Applications in Linear Algebra. Russ Math. 67, 74–81 (2023). https://doi.org/10.3103/S1066369X23010012
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DOI: https://doi.org/10.3103/S1066369X23010012