Automorphisms of the endomorphism semigroup of a polynomial algebra

A. Belov-Kanel, R. Lipyanski

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We describe the automorphism group of the endomorphism semigroup End(K[x1,...,xn]) of ring K[x1,...,xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15].More precisely, we prove that if. AutEnd(K[x1,...,xn]) then there exists a semi-linear automorphism s:K[x1,...,xn]K[x1,...,xn] such that (g)=sgs1 for any gEnd(K[x1,...,xn]). This extends the result obtained by A. Berzins for an infinite field K.

Original languageEnglish
Pages (from-to)40-54
Number of pages15
JournalJournal of Algebra
Issue number1
StatePublished - 1 May 2011

Bibliographical note

Funding Information:
The authors are grateful to B. Plotkin for attracting their attention to this problem and interest to this work. The first author was supported by the Israel Science Foundation (grant No. 1178/06).


  • Kronecker endomorphism
  • Polynomial algebra
  • Rank endomorphism
  • Semi-inner automorphism
  • Variety of commutative-associative algebras


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