-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

DOI:
https://doi.org/10.1090/S0002-9939-99-04530-X

MathSciNet review:
1468180

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

**[A1]**S. A. Amitsur,*Rings with involution*, Israel J. Math.**6**(1968), 99 - 106. MR**39:256****[A2]**S. A. Amitsur,*Identities in rings with involution*, Israel J. Math.**7**(1969), 63 - 68. MR**39:4216****[BGR]**Y. Bahturin, A. Giambruno and D. Riley,*Group-graded algebras with polynomial identity*, Israel J. Math.**104**(1998), 145-156.**[BZ]**Y. Bahturin and M. Zaicev,*Identities of graded algebras*, J. Algebra, to appear.**[GR]**A. Giambruno and A. Regev,*Wreath products and P.I. algebras*, J. Pure Applied Algebra**35**(1985), 133 -149. MR**86e:16027****[K]**V. K. Kharchenko,*Galois extensions and quotient rings*, Algebra i Logika**13**(1974), 460 - 484 (Russian); English transl. (1975), 264 - 281. MR**53:498****[L]**V. N. Latyshev,*On the theorem of Regev about identities in the tensor product of P.I. algebras*, Uspekhi Mat. Nauk.**27**(1972), 213 - 214 (Russian). MR**52:13924****[M]**S. Montgomery,*Fixed rings of finite automorphism groups of associative rings*, LNM n. 818 Springer-Verlag, Berlin, 1980. MR**81j:16041****[R]**Yu. P. Razmyslov,*Identities of Algebras and Their Representations*, Transl. Math. Monogr., vol. 138, Amer. Math. Soc., Providence RI, 1994 xiii+318pp. MR**95i:16022****[Re1]**A. Regev,*Existence of identities in*, Israel J. Math.**11**(1972), 131 - 152. MR**47:3442****[Re2]**A. Regev,*The representations of and explicit identities for P.I. algebras*, J. Algebra**51**(1978), 25 - 40. MR**57:9745**

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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

Email:
bahturin@mech.math.msu.su

**A. Giambruno**

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

Email:
a.giambruno@unipa.it

**M. Zaicev**

Email:
zaicev@nw.math.msu.su

DOI:
https://doi.org/10.1090/S0002-9939-99-04530-X

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