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

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

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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\by A.~Ya.~Belov
\paper No associative $PI$-algebra coincides with its commutant
\jour Sibirsk. Mat. Zh.
\yr 2003
\vol 44
\issue 6
\pages 1239--1254
\mathnet{http://mi.mathnet.ru/smj1251}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2034931}
\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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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
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
3. L. M. Samoǐlov, “The unitary closure property of the prime varieties of associative algebras”, Siberian Math. J., 51:4 (2010), 712–722
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
5. Alves Jorge S.M., “On Graded Central Polynomials of the Graded Algebra M-2(E)”, Comm Algebra, 38:6 (2010), 2184–2198
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
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
8. L. M. Samoilov, “On the Primality Property of Central Polynomials of Prime Varieties of Associative Algebras”, Math. Notes, 99:3 (2016), 413–416
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