No associative PI-algebra coincides with its commutant

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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).

Original languageEnglish
Pages (from-to)969-980
Number of pages12
JournalSiberian Mathematical Journal
Issue number6
StatePublished - 1 Nov 2003
Externally publishedYes


  • Capelli identity
  • Central polynomial
  • Forms
  • Hamilton-Cayley equation
  • Identity
  • Identity with trace
  • PI-algebra
  • Stable variety
  • T-prime variety
  • Variety of algebras
  • Weak identity


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