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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Automorphisms of a free associative algebra of rank $ 2$. II


Author: Anastasia J. Czerniakiewicz
Journal: Trans. Amer. Math. Soc. 171 (1972), 309-315
MSC: Primary 16A72
MathSciNet review: 0310021
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Abstract: Let $ R$ be a commutative domain with 1. $ R\langle x,y\rangle $ stands for the free associative algebra of rank 2 over $ R;R[\tilde x,\tilde y]$ is the polynomial algebra over $ R$ in the commuting indeterminates $ \tilde x$ and $ \tilde y$.

We prove that the map Ab$ : \operatorname{Aut} (R\langle x,y\rangle ) \to \operatorname{Aut} (R[\tilde x,\tilde y])$ induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of $ F[\tilde x,\tilde y]$ ($ F$ a field) we are able to conclude that every automorphism of $ F\langle x,y\rangle $ is tame (i.e. a product of elementary automorphisms).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0310021-2
PII: S 0002-9947(1972)0310021-2
Keywords: Free associative algebra, endomorphisms, automorphisms, elementary automorphisms, tame automorphisms, wild automorphisms, polynomial rings, euclidean domains
Article copyright: © Copyright 1972 American Mathematical Society