A polynomial identity in a ring is a noncommutative polynomial in a finite number of variables which evaluates to zero for any choice substituted in place of the variables.
If is an algebra in characteristic then is an identity in ; this is a rather trivial case. It is a very special property of a ring to admit a nontrivial polynomial identity.
A polynomial identity ring is a ring with at least one polynomial identity with at least one of the coefficients .
The commutativity identity holds in any commutative ring.
(Amitsur–Levitski theorem) In the ring of -matrices over any field the Amitsur–Levitzki standard identity holds,
where is the symmetric group on letters. In addition, no monic polynomial identity in degree less than exists for .
(Artin1969) Let be a prime field of characteristic . A ring which is also a -algebra is an Azumaya algebra of rank over its zenter iff the two conditions hold:
(i) it does not have a representation over a field of dimension strictly less than
(ii) it satisfies all polynomial identities which hold in the ring of -matrices in characteristic
Irving Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948) 575–580
S. A. Amitsur, Polynomial identities, Israel J. Math 19 (1974) 183–199
S. A. Amitsur, Associative rings with identities. In: I. N. Herstein (eds) Some Aspects of Ring Theory. C.I.M.E. Summer Schools 37. Springer 2010 doi
J. Levitzki, A theorem on polynomial identities, Proc. Amer. Math. Soc. 1 (1950) 449–463
Vesselin Drensky, Edward Formanek, Polynomial identity rings, Birkhauser (2004) viii+200 pp [doi:10.1007/978-3-0348-7934-7]
review:
Louis Rowen, Bull. AMS 43 2 (2006) 259–267 [pdf, doi:2006-43-02/S0273-0979-06-01082-2]
Wikipedia: Polynomial identity ring, Amitsur–Levitski theorem
Michael Artin, On Azumaya algebras and finite-dimensional representations of rings, J. Algebra 11 (1969) 532–563 [doi:10.1016/0021-8693(69)90091]
Claudio Procesi, On the theorem of Amitsur–Levitzki, Israel Journal of Mathematics 207: 151—154 (arXiv:1308.2421 doi)
Last revised on September 4, 2024 at 09:11:51. See the history of this page for a list of all contributions to it.