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 -