Automorphisms of a free associative algebra of rank $2$. I
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- by Anastasia J. Czerniakiewicz
- Trans. Amer. Math. Soc. 160 (1971), 393-401
- DOI: https://doi.org/10.1090/S0002-9947-1971-0280549-1
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Abstract:
Let ${R_{\left \langle 2 \right \rangle }} = R\left \langle {x,y} \right \rangle$ be the free associative algebra of rank 2, on the free generators $x$ and $y$, over $R$ ($R$ a field, a Euclidean domain, etc.). We prove that if $\varphi$ is an automorphism of ${R_{\left \langle 2 \right \rangle }}$ that keeps $(xy - yx)$ fixed (up to multiplication by an element of $R$), then $\varphi$ is tame, i.e. it is a product of elementary automorphisms of ${R_{\left \langle 2 \right \rangle }}$. This follows from a more general result about endomorphisms of ${R_{\left \langle 2 \right \rangle }}$ via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2.References
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Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 160 (1971), 393-401
- MSC: Primary 16.60
- DOI: https://doi.org/10.1090/S0002-9947-1971-0280549-1
- MathSciNet review: 0280549