The Braun–Kemer–Razmyslov Theorem for affine PI-algebras

Translated title of the contribution: The Braun–Kemer–Razmyslov Theorem for affine PI-algebras

Alexei Kanel Belov, Louis Rowen

Research output: Contribution to journalArticlepeer-review


A self-contained, combinatoric exposition is given for the Braun–Kemer–Razmyslov Theorem over an arbitrary commutative Noetherian ring.At one time, the community did not believe in the validity of this result, and contrary to public opinion, the corresponding question was posed by V.N. Latyshev in his doctoral dissertation. One of the major theorems in the theory of PI algebras is the Braun-Kemer-Razmyslov Theorem. We preface its statement with some basic definitions. 1. An algebra A is affine over a commutative ring C if A is generated as an algebra over C by a finite number of elements α1, . . ., αℓ; in this case we write A= C{ α1, . . ., a}. We say the algebra A is finite if A is spanned as a C-module by finitely many elements. 2. Algebras over a field are called PI algebras if they satisfy (nontrivial) polynomial identities. 3. The Capelli polynomial Capk of degree 2k is defined as (Equation presented) 4. Jac(A) denotes the Jacobson radical of the algebra A which, for PI-algebras is the intersection of the maximal ideals of A, in view of Kaplansky’s theorem. The aim of this article is to present a readable combinatoric proof of the theorem: The Braun-Kemer-Razmyslov Theorem The Jacobson radical Jac(A) of any affine PI algebra A over a field is nilpotent.

Translated title of the contributionThe Braun–Kemer–Razmyslov Theorem for affine PI-algebras
Original languageEnglish
Pages (from-to)89-128
Number of pages40
JournalChebyshevskii Sbornik
Issue number3
StatePublished - 2020


  • Algebras with polynomial identity
  • Hilbert series
  • Relatively free algebras
  • Representable algebras
  • Specht problem
  • Varieties of algebras

All Science Journal Classification (ASJC) codes

  • General Mathematics


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