Polynomial identity ring

${\displaystyle P(r_1,r_2,\dots ,r_N)=0}$

for all N-tuples r₁, r₂, ..., r_N taken from R.

Strictly the X_i here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra.

If the degree of the polynomial P is defined in the usual way, the polynomial P is called monic if at least one of its terms of highest degree has coefficient equal to 1.

Every commutative ring is a PI-ring, satisfying the polynomial identity XY − YX = 0. Therefore, PI-rings are usually taken as close generalizations of commutative rings. If the ring has characteristic p different from zero then it satisfies the polynomial identity pX = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.

Examples

• For example, if R is a commutative ring it is a PI-ring: this is true with

${\displaystyle P(X_1,X_2)=X_1{X}_2-X_2{X}_1=0}$

• The ring of 2 × 2 matrices over a commutative ring satisfies the Hall identity

${\displaystyle (xy-yx{)}^{2}z=z(xy-yx{)}^2}$

This identity was used by M. Hall (1943), but was found earlier by Wagner (1937).

• A major role is played in the theory by the standard identity s_N, of length N, which generalises the example given for commutative rings (N = 2). It derives from the Leibniz formula for determinants

${\displaystyle det(A)=\sum _{\sigma \in S_N}\mathrm{sgn}(\sigma )\prod _{i=1}^N{a}_{i,\sigma (i)}}$

by replacing each product in the summand by the product of the X_i in the order given by the permutation σ. In other words each of the N ! orders is summed, and the coefficient is 1 or −1 according to the signature.

${\displaystyle s_N(X_1,\dots ,X_N)=\sum _{\sigma \in S_N}\mathrm{sgn}(\sigma )X_{\sigma (1)}\cdots X_{\sigma (N)}=0}$

The m × m matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies s_2m. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2m.

• Given a field k of characteristic zero, take R to be the exterior algebra over a countably infinite-dimensional vector space with basis e₁, e₂, e₃, ... Then R is generated by the elements of this basis and

e_i e_j = − e_j e_i.

This ring does not satisfy s_N for any N and therefore can not be embedded in any matrix ring. In fact s_N(e₁,e₂,...,e_N) = N ! e₁e₂...e_N ≠ 0. On the other hand it is a PI-ring since it satisfies [[x, y], z] := xyz − yxz − zxy + zyx = 0. It is enough to check this for monomials in the e_i's. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree [x, y] := xy − yx = 0. If both are of odd degree then [x, y] = xy − yx = 2xy has even degree and therefore commutes with z, i.e. [[x, y], z] = 0.