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Mat. Sb., 2009, Volume 200, Number 12, Pages 41–62 (Mi msb7531)  

This article is cited in 5 scientific papers (total in 5 papers)

Upper bound for the length of commutative algebras

O. V. Markova

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field one means the least positive integer $k$ such that the words of length not exceeding $k$ span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, an upper bound for the length of a commutative algebra in terms of a function of two invariants of the algebra, the dimension and the maximal degree of the minimal polynomial for the elements of the algebra, is obtained. As a corollary, a formula for the length of the algebra of diagonal matrices over an arbitrary field is obtained.
Bibliography: 8 titles.

Keywords: length of an algebra, matrix theory, commutative algebra, algebra of diagonal matrices.

DOI: https://doi.org/10.4213/sm7531

Full text: PDF file (529 kB)
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English version:
Sbornik: Mathematics, 2009, 200:12, 1767–1787

Bibliographic databases:

UDC: 512.552
MSC: Primary 16P10; Secondary 16R20
Received: 28.01.2009

Citation: O. V. Markova, “Upper bound for the length of commutative algebras”, Mat. Sb., 200:12 (2009), 41–62; Sb. Math., 200:12 (2009), 1767–1787

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. O. V. Markova, “The length function and matrix algebras”, J. Math. Sci., 193:5 (2013), 687–768  mathnet  crossref
    2. O. V. Markova, “On the Relationship between the Length of an Algebra and the Index of Nilpotency of Its Jacobson Radical”, Math. Notes, 94:5 (2013), 636–641  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Guterman A.E., Markova O.V., Mehrmann V., “Lengths of quasi-commutative pairs of matrices”, Linear Alg. Appl., 498:SI (2016), 450–470  crossref  mathscinet  zmath  isi  scopus
    4. A. E. Guterman, O. V. Markova, “Dlina gruppovykh algebr grupp nebolshogo razmera”, Chislennye metody i voprosy organizatsii vychislenii. XXXI, Zap. nauchn. sem. POMI, 472, POMI, SPb., 2018, 76–87  mathnet
    5. N. A. Kolegov, O. V. Markova, “Sistemy porozhdayuschikh matrichnykh algebr intsidentnosti nad konechnymi polyami”, Chislennye metody i voprosy organizatsii vychislenii. XXXI, Zap. nauchn. sem. POMI, 472, POMI, SPb., 2018, 120–144  mathnet
  • Математический сборник Sbornik: Mathematics (from 1967)
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