polynomial identity algebra

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 $x_{i}^{\prime}s\in X$. 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. 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 $x^{q}-x$.

3. 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. 4.

The ring $S$ of $2\times 2$ 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. 5.

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

6. 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 $R$.

Title polynomial identity algebra PolynomialIdentityAlgebra 2013-03-22 14:20:38 2013-03-22 14:20:38 CWoo (3771) CWoo (3771) 11 CWoo (3771) Definition msc 16U80 msc 16R10 PI-algebra algebra with polynomial identity Hall identity