PolynomialIdentityAlgebra 2004-04-29 15:15:47 2007-03-22 14:23:58 Definition Hall identity % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $R$ be a commutative ring with 1. Let $X$ be a countable set of \emph{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 $X$. \textbf{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 \emph{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 \emph{polynomial identity for} $A$. A polynomial $f\in R\langle X\rangle$ is \emph{proper}, or \emph{monic}, if, in the homogeneous component of the highest degree in $f$, one of its monomials has coefficient = 1. \textbf{Definition}. An algebra $A$ over a commutative ring $R$ is said to be a \emph{polynomial identity algebra over} $R$, or a PI-\emph{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 \emph{polynomial identity ring}, or PI-\emph{ring}, $R$ is a polynomial identity $\mathbb{Z}$-algebra. \textbf{Examples} \begin{enumerate} \item A commutative ring is a PI-ring, satisfying the polynomial $[x,y]=xy-yx$. \item A finite field (with $q$ elements) is a PI-ring, satisfying $x^q-x$. \item 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 $[x_1,y_1][x_2,y_2]\cdots [x_n,y_n]$. \item 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 \emph{Hall identity}. \item A subring of a PI-ring is a PI-ring. A homomorphic image of a PI-ring is a PI-ring. \item 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 $R$. \end{enumerate}