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

Definition. Let be a -algebra and . For any , is called an evaluation of at -tuple . If the evaluation vanishes (=0) for all -tuples of , then is called a polynomial identity for .

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

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

Examples

1. A commutative ring is a PI-ring, satisfying the polynomial .
2. A finite field (with elements) is a PI-ring, satisfying .
3. The ring of upper triangular matrices over a field is a PI-ring. This is true because for any , is strictly upper triangular (zeros along the diagonal). Any product of strictly upper triangular matrices in is 0. Therefore, satisfies .
4. The ring of matrices over a field is a PI-ring. One can show that satisfies . This identity is called the Hall identity.
5. A subring of a PI-ring is a PI-ring. A homomorphic image of a PI-ring is a PI-ring.
6. One can show that a ring with polynomial identity is commutative. Thus, one sees that and , although very different (one is homogeneous of degree 2 in 2 variables, the other one is not even homogeneous, in one variable of degree n), are both polynomial identities for .