Automorphisms of a free associative algebra of rank $2$. II
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- by Anastasia J. Czerniakiewicz PDF
- Trans. Amer. Math. Soc. 171 (1972), 309-315 Request permission
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 $\text {Ab}: \operatorname {Aut} (R\langle x,y\rangle ) \to \operatorname {Aut} (R[\tilde x,\tilde y])$ 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).References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 171 (1972), 309-315
- MSC: Primary 16A72
- DOI: https://doi.org/10.1090/S0002-9947-1972-0310021-2
- MathSciNet review: 0310021