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$G$-identities on associative algebras

Authors: Y. Bahturin, A. Giambruno and M. Zaicev
Journal: Proc. Amer. Math. Soc. 127 (1999), 63-69
MSC (1991): Primary 16R50; Secondary 16W20
MathSciNet review: 1468180
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Abstract: Let $R$ be an algebra over a field and $G$ a finite group of automorphisms and anti-automorphisms of $R$. We prove that if $R$ satisfies an essential $G$-polynomial identity of degree $d$, then the $G$-codimensions of $R$ are exponentially bounded and $R$ satisfies a polynomial identity whose degree is bounded by an explicit function of $d$. As a consequence we show that if $R$ is an algebra with involution $*$ satisfying a $*$-polynomial identity of degree $d$, then the $*$-codimensions of $R$ are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case $R$ must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

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Additional Information

Y. Bahturin
Affiliation: $\mathrm{(Y. Bahturin and M. Zaicev)}$ Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119899 Russia

A. Giambruno
Affiliation: $\mathrm{(A. Giambruno)}$ Dipartimento di Matematica e Applicazioni, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy

M. Zaicev

Received by editor(s): December 18, 1996
Received by editor(s) in revised form: May 13, 1997
Additional Notes: Y. Bahturin and M. Zaicev acknowledge support by the Russian Foundation of Fundamental Research, grant 96-01-00146. A. Giambruno was supported by MURST and CNR of Italy.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1999 American Mathematical Society