standard identity

Let R be a commutative ring and X be a set of non-commuting variablesMathworldPlanetmath over R. The standard identity of degree n in RX, denoted by [x1,xn], is the polynomialMathworldPlanetmathPlanetmath

πsign(π)xπ(1)xπ(n), where πSn.


  • A ring R satisfying the standard identity of degree 2 (i.e., [R,R]=0) is commutativePlanetmathPlanetmathPlanetmath. In this sense, algebrasMathworldPlanetmathPlanetmathPlanetmath satisfying a standard identity is a generalizationPlanetmathPlanetmath of the class of commutative algebras.

  • Two immediate properties of [x1,xn] are that it is multilinear over R, and it is alternating, in the sense that [r1,rn]=0 whenever two of the ris are equal. Because of these two properties, one can show that an n-dimensional algebra R over a field k is a PI-algebra, satisfying the standard identity of degree n+1. As a corollary, 𝕄n(k), the n×n matrix ring over a field k, is a PI-algebra satisfying the standard identity of degree n2+1. In fact, Amitsur and Levitski have shown that 𝕄n(k) actually satisfies the standard identity of degree 2n.


  • 1 S. A. Amitsur and J. Levitski, MinimalPlanetmathPlanetmath identitiesPlanetmathPlanetmathPlanetmathPlanetmath for algebras, Proc. Amer. Math. Soc., 1 (1950) 449-463.
Title standard identity
Canonical name StandardIdentity
Date of creation 2013-03-22 14:21:10
Last modified on 2013-03-22 14:21:10
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 16R10