TY - JOUR

T1 - Automorphisms of the endomorphism semigroup of a polynomial algebra

AU - Belov-Kanel, A.

AU - Lipyanski, R.

N1 - 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).

PY - 2011/5/1

Y1 - 2011/5/1

N2 - 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.

AB - 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.

KW - Kronecker endomorphism

KW - Polynomial algebra

KW - Rank endomorphism

KW - Semi-inner automorphism

KW - Variety of commutative-associative algebras

UR - http://www.scopus.com/inward/record.url?scp=79953044029&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2011.01.020

DO - 10.1016/j.jalgebra.2011.01.020

M3 - Article

AN - SCOPUS:79953044029

SN - 0021-8693

VL - 333

SP - 40

EP - 54

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -