nLab polynomial identity ring

Contents

Idea

A polynomial identity in a ring RR is a noncommutative polynomial in a finite number of variables x 1,,x nx_1,\ldots,x_n which evaluates to zero for any choice a 1,a na_1\ldots,a_n substituted in place of the variables.

If RR is an algebra in characteristic pp then pxp x is an identity in RR; 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 ±1\pm 1.

Examples

  • The commutativity identity x 1x 2x 2x 1x_1 x_2 - x_2 x_1 holds in any commutative ring.

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

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

    where Σ(2n)\Sigma(2n) is the symmetric group on 2n2n letters. In addition, no monic polynomial identity in degree less than 2n2n exists for M n(F)M_n(F).

Properties

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

(i) it does not have a representation over a field KkK\supset k of dimension strictly less than nn

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

Literature

category: algebra

Last revised on September 4, 2024 at 09:11:51. See the history of this page for a list of all contributions to it.