nLab polynomial identity ring

Idea

A polynomial identity in a ring $R$ is a noncommutative polynomial in a finite number of variables $x_1,\ldots,x_n$ which evaluates to zero for any choice $a_1\ldots,a_n$ substituted in place of the variables.

If $R$ is an algebra in characteristic $p$ then $p x$ is an identity in $R$; this is a rather trivial case. It is a very special property of a ring to admit a nontrivial polynomial identity.

Definition

A polynomial identity ring is a ring with at least one polynomial identity with at least one of the coefficients $\pm 1$.

Examples

• The commutativity identity $x_1 x_2 - x_2 x_1$ holds in any commutative ring.

• (Amitsur–Levitski theorem) In the ring $M_n(F)$ of $n\times n$-matrices over any field $F$ the Amitsur–Levitzki standard identity holds,

  $$\sum_{\pi\in\Sigma(2 n)} sign(\pi) x_{\pi(1)}\cdots x_{\pi(2n)},$$

  where $\Sigma(2n)$ is the symmetric group on $2n$ letters. In addition, no monic polynomial identity in degree less than $2n$ exists for $M_n(F)$.

Properties

(Artin1969) Let $k$ be a prime field of characteristic $p$. A ring $A$ which is also a $k$-algebra is an Azumaya algebra of rank $n^2$ over its zenter $Z(A)$ iff the two conditions hold:

(i) it does not have a representation over a field $K\supset k$ of dimension strictly less than $n$

(ii) it satisfies all polynomial identities which hold in the ring of $n\times n$-matrices in characteristic $p$

Literature

• 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)

category: algebra