polynomial identity algebra
Let $R$ be a commutative ring with 1. Let $X$ be a countable set of variables, and let $R\u27e8X\u27e9$ denote the free associative algebra over $R$. If $X$ is finite, we can also write $R\u27e8X\u27e9$ as $R\u27e8{x}_{1},\mathrm{\dots}{x}_{n}\u27e9$, where the ${x}_{i}^{\prime}s\in X$. Because of the freeness condition on the algebra^{}, the variables are noncommuting among themselves. However, the variables do commute with elements of $R$. A typical element $f$ of $R\u27e8X\u27e9$ is a polynomial^{} over $R$ in $n$ (finite) noncommuting variables of $X$.
Definition. Let $A$ be a $R$algebra and $f=f({x}_{1},\mathrm{\dots},{x}_{n})\in R\u27e8X\u27e9$. For any ${a}_{1},\mathrm{\dots},{a}_{n}\in A$, $f({a}_{1},\mathrm{\dots},{a}_{n})\in A$ is called an evaluation of $f$ at $n$tuple $\mathrm{(}{a}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{a}_{n}\mathrm{)}$. If the evaluation vanishes (=0) for all $n$tuples of ${\mathrm{\Pi}}_{i=1}^{n}A$, then $f$ is called a polynomial identity for $A$.
A polynomial $f\in R\u27e8X\u27e9$ 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 PIalgebra over $R$, if there is a proper polynomial $f\in R\u27e8{x}_{1},\mathrm{\dots},{x}_{n}\u27e9$, such that $f$ is a polynomial identity for $A$. A polynomial identity ring, or PIring, $R$ is a polynomial identity $\mathbb{Z}$algebra.
Examples

1.
A commutative ring is a PIring, satisfying the polynomial $[x,y]=xyyx$.

2.
A finite field (with $q$ elements) is a PIring, satisfying ${x}^{q}x$.

3.
The ring $T$ of upper triangular $n\times n$ matrices over a field is a PIring. This is true because for any $a,b\in T$, $abba$ is strictly upper triangular (zeros along the diagonal). Any product^{} of $n$ strictly upper triangular matrices^{} in $T$ is 0. Therefore, $T$ satisfies $[{x}_{1},{y}_{1}][{x}_{2},{y}_{2}]\mathrm{\cdots}[{x}_{n},{y}_{n}]$.

4.
The ring $S$ of $2\times 2$ matrices over a field is a PIring. 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 PIring is a PIring. A homomorphic image^{} of a PIring is a PIring.

6.
One can show that a ring $R$ with polynomial identity ${x}^{n}x$ is commutative^{}. Thus, one sees that ${x}^{n}x$ and $xyyx$, 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 

Canonical name  PolynomialIdentityAlgebra 
Date of creation  20130322 14:20:38 
Last modified on  20130322 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  PIalgebra 
Synonym  algebra with polynomial identity 
Defines  Hall identity 