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Sibirsk. Mat. Zh., 2003, Volume 44, Number 6, Pages 1239–1254 (Mi smj1251)  

This article is cited in 8 scientific papers (total in 8 papers)

No associative $PI$-algebra coincides with its commutant

A. Ya. Belovab

a Moscow Institute of Open Education
b International University Bremen

Abstract: We prove that each (possibly not finitely generated) associative $PI$-algebra does not coincide with its commutant. We thus solve I. V. L'vov's problem in the Dniester Notebook. The result follows from the fact (also established in this article) that, in every $T$-prime variety, some weak identity holds and there exists a central polynomial (the existence of a central polynomial was earlier proved by A. R. Kemer). Moreover, we prove stability of $T$-prime varieties (in the case of characteristic zero, this was done by S. V. Okhitin who used A. R. Kemer's classification of $T$-prime varieties).

Keywords: PI-algebra, variety of algebras, identity, stable variety, weak identity, identity with trace, forms, Capelli identity, T-prime variety, Hamilton?Cayley equation, central polynomial

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English version:
Siberian Mathematical Journal, 2003, 44:6, 969–980

Bibliographic databases:

UDC: 512.552.4, 512.554.32, 512.664.2
Received: 12.05.2003

Citation: A. Ya. Belov, “No associative $PI$-algebra coincides with its commutant”, Sibirsk. Mat. Zh., 44:6 (2003), 1239–1254; Siberian Math. J., 44:6 (2003), 969–980

Citation in format AMSBIB
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\paper No associative $PI$-algebra coincides with its commutant
\jour Sibirsk. Mat. Zh.
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\vol 44
\issue 6
\pages 1239--1254
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\zmath{https://zbmath.org/?q=an:1054.16015}
\elib{http://elibrary.ru/item.asp?id=5219458}
\transl
\jour Siberian Math. J.
\yr 2003
\vol 44
\issue 6
\pages 969--980
\crossref{https://doi.org/10.1023/B:SIMJ.0000007472.85188.56}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Brandao Antonio Pereira J., Koshlukov P., “Central polynomials for Z(2)–graded algebras and for algebras with involution”, Journal of Pure and Applied Algebra, 208:3 (2007), 877–886  crossref  mathscinet  zmath  isi  scopus
    2. L. M. Samoilov, “On Multilinear Components of Prime Subvarieties in the Variety $\mathrm{Var}(M_{1,1})$”, Math. Notes, 87:6 (2010), 890–902  mathnet  crossref  crossref  mathscinet  isi
    3. L. M. Samoǐlov, “The unitary closure property of the prime varieties of associative algebras”, Siberian Math. J., 51:4 (2010), 712–722  mathnet  crossref  mathscinet  isi  elib  elib
    4. Brandao Antonio Pereira J., Koshlukov P., Krasilnikov A., da Silva E.A., “The Central Polynomials for the Grassmann Algebra”, Israel J Math, 179:1 (2010), 127–144  crossref  mathscinet  zmath  isi  elib  scopus
    5. Alves Jorge S.M., “On Graded Central Polynomials of the Graded Algebra M-2(E)”, Comm Algebra, 38:6 (2010), 2184–2198  crossref  mathscinet  zmath  isi  scopus
    6. A. R. Chekhlov, Ml. V. Agafontseva, “Ob abelevykh gruppakh s tsentralnymi kvadratami kommutatorov endomorfizmov”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2013, no. 4(24), 54–59  mathnet  elib
    7. Pereira da Silva e Silva D.D., “a Primeness Property For Central Polynomials of Verbally Prime Pi Algebras”, Linear Multilinear Algebra, 63:11 (2015), 2151–2158  crossref  mathscinet  zmath  isi  scopus
    8. L. M. Samoilov, “On the Primality Property of Central Polynomials of Prime Varieties of Associative Algebras”, Math. Notes, 99:3 (2016), 413–416  mathnet  crossref  crossref  mathscinet  isi  elib
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