polynomial identity algebra

Let R be a commutative ring with 1. Let X be a countable set of variables, and let RX denote the free associative algebra over R. If X is finite, we can also write RX as Rx1,xn, where the xisX. Because of the freeness condition on the algebraMathworldPlanetmathPlanetmath, the variables are non-commuting among themselves. However, the variables do commute with elements of R. A typical element f of RX is a polynomialMathworldPlanetmathPlanetmath over R in n (finite) non-commuting variables of X.

Definition. Let A be a R-algebra and f=f(x1,,xn)RX. For any a1,,anA, f(a1,,an)A is called an evaluation of f at n-tuple (a1,,an). If the evaluation vanishes (=0) for all n-tuples of Πi=1nA, then f is called a polynomial identity for A.

A polynomial fRX is proper, or monic, if, in the homogeneous component of the highest degree in f, one of its monomials has coefficient = 1.

Definition. An algebra A over a commutative ring R is said to be a polynomial identity algebra over R, or a PI-algebra over R, if there is a proper polynomial fRx1,,xn, such that f is a polynomial identity for A. A polynomial identity ring, or PI-ring, R is a polynomial identity -algebra.


  1. 1.

    A commutative ring is a PI-ring, satisfying the polynomial [x,y]=xy-yx.

  2. 2.

    A finite field (with q elements) is a PI-ring, satisfying xq-x.

  3. 3.

    The ring T of upper triangular n×n matrices over a field is a PI-ring. This is true because for any a,bT, ab-ba is strictly upper triangular (zeros along the diagonal). Any productMathworldPlanetmathPlanetmath of n strictly upper triangular matricesMathworldPlanetmath in T is 0. Therefore, T satisfies [x1,y1][x2,y2][xn,yn].

  4. 4.

    The ring S of 2×2 matrices over a field is a PI-ring. One can show that S satisfies [[x1,x2]2,x3]. This identityPlanetmathPlanetmathPlanetmathPlanetmath is called the Hall identity.

  5. 5.

    A subring of a PI-ring is a PI-ring. A homomorphic imagePlanetmathPlanetmathPlanetmath of a PI-ring is a PI-ring.

  6. 6.

    One can show that a ring R with polynomial identity xn-x is commutativePlanetmathPlanetmathPlanetmath. Thus, one sees that xn-x and xy-yx, although very different (one is homogeneous of degree 2 in 2 variables, the other one is not even homogeneousPlanetmathPlanetmathPlanetmathPlanetmath, in one variable of degree n), are both polynomial identities for R.

Title polynomial identity algebra
Canonical name PolynomialIdentityAlgebra
Date of creation 2013-03-22 14:20:38
Last modified on 2013-03-22 14:20:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 16U80
Classification msc 16R10
Synonym PI-algebra
Synonym algebra with polynomial identity
Defines Hall identity