Let $R$ be a commutative ring with 1. Let $X$ be a countable set of variables, and let $R \langle X \rangle$ denote the free associative algebra over $R$ . If $X$ is finite, we can also write $R \langle X \rangle$ as $R \langle x_1, \ldots\, x_n \rangle$ , where the $x_i's \in X$ . Because of the freeness condition on the algebra, the variables are non-commuting among themselves. However, the variables do commute with elements of $R$ . A typical element $f$ of $R\langle X\rangle$ is a polynomial over $R$ in $n$ (finite) non-commuting variables of $X$ .

Definition. Let $A$ be a $R$ -algebra and $f=f(x_1,\ldots,x_n)\in R\langle X\rangle$ . For any $a_1,\ldots,a_n\in A$ , $f(a_1,\ldots,a_n)\in A$ is called an evaluation of $f$ at $n$ -tuple $(a_1,\ldots,a_n)$ . If the evaluation vanishes (=0) for all $n$ -tuples of $\Pi_{i=1}^{n}A$ , then $f$ is called a polynomial identity for $A$ .

A polynomial $f\in R\langle X\rangle$ 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 $f \in R \langle x_1, \ldots, x_n \rangle$ , such that $f$ is a polynomial identity for $A$ . A polynomial identity ring, or PI-ring, $R$ is a polynomial identity $\mathbb{Z}$ -algebra.

Examples

1. A commutative ring is a PI-ring, satisfying the polynomial $[x,y]=xy-yx$ .
2. A finite field (with $q$ elements) is a PI-ring, satisfying $x^q-x$ .
3. The ring $T$ of upper triangular $n \times n$ matrices over a field is a PI-ring. This is true because for any $a, b\in T$ , $ab-ba$ is strictly upper triangular (zeros along the diagonal). Any product of $n$ strictly upper triangular matrices in $T$ is 0. Therefore, $T$ satisfies $[x_1,y_1][x_2,y_2]\cdots [x_n,y_n]$ .
4. The ring $S$ of $2\times2$ matrices over a field is a PI-ring. One can show that $S$ satisfies $[[x_1,x_2]^2,x_3]$ . 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 $R$ with polynomial identity $x^n-x$ is commutative. Thus, one sees that $x^n-x$ and $xy-yx$ , 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 $R$ .