polynomial identity algebra

Let $R$ be a commutative ring with 1. Let $X$ be a countable set of variables, and let $R⟨X⟩$ denote the free associative algebra over $R$. If $X$ is finite, we can also write $R⟨X⟩$ as $R⟨x_1,\mathrm{\dots }{x}_n⟩$, 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 $R⟨X⟩$ is a polynomial over $R$ in $n$ (finite) non-commuting variables of $X$.

Definition. Let $A$ be a $R$-algebra and $f=f(x_1,\mathrm{\dots },x_n)\in R⟨X⟩$. 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 $f\in R⟨X⟩$ 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 $f\in R⟨x_1,\mathrm{\dots },x_n⟩$, 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 $[x_1,y_1][x_2,y_2]\mathrm{\cdots }[x_n,y_n]$.

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. 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 $R$.

Title	polynomial identity algebra
Canonical name	PolynomialIdentityAlgebra
Date of creation	2013-03-22 14:20:38
Last modified on	2013-03-22 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	PI-algebra
Synonym	algebra with polynomial identity
Defines	Hall identity