Abstract
An example is given of a ringR (with 1) satisfying the standard identityS 6[x 1, ...,x 6] butM 2(R), the 2 × 2 matrix ring overR, does not satisfyS 12[x 1, ...,x 12]. This is in contrast to the caseR=M n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS 2n [x 1, ...,x 2n] andM 2(R) satisfiesS 4n [x 1, ...,x n].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Braun,On a question of G. Bergman and L. Small on Azumaya algebras, J. Algebra87 (1984), 24–45.
F. Demeyer and E. Ingraham,Separable Algebras over Commutative Rings, Lecture Notes in Math.181, Springer-Verlag, Berlin and New York, 1971.
I. Kaplansky,Fields and Rings, University of Chicago Press, Chicago, 1970.
U. Leron,Nil and power central polynomial in rings, Trans. Amer. Math. Soc.202 (1975), 97–103.
C. Procesi,Rings with Polynomial Identities, Dekker, New York, 1973.
A. Regev,The Kronecker product of S n-characters and an A × B theorem for Capelli identities, J. Algebra66 (1980), 505–510.
L. H. Rowen,Polynomial Identities in Ring Theory, Academic Press, New York, 1980.
Author information
Authors and Affiliations
Additional information
Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University of Texas at Austin and the University of Washington at Seattle.
Rights and permissions
About this article
Cite this article
Braun, A. On P.I. ring and standard identities. Israel J. Math. 55, 345–349 (1986). https://doi.org/10.1007/BF02765031
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02765031