We will assume that is a commutative ring with 1. Also, given an integer we will denote by the ring of polynomials in non-commuting variables and with coefficients in
Definition. A -algebra is called a PI-algebra if there exists an integer and a non-zero polynomial
such that the coefficient of at least one of the monomials of the highest degree in is 1 and for all Then the smallest possible degree of such is called the PI-degree of
Remark 1. If an algebra satisfies a polynomial then obviously every subring of and homomorphic image of will satisfy too.
Remark 2. It is known and easy to prove that if an algebra satisfies a non-zero polynomial of degree then will also satisfy a non-zero multi-linear polynomial of degree at most
Example 1. Every commutative algebra is PI because it satisfies the polynomial
Example 2. Let be a commutative ring and Then for any we have Thus by Cayley-Hamilton theorem for some Therefore commutes with any element of So we’ve proved that is PI because it satisfies the polynomial
Remark 2. Let be a commutative ring and suppose that satisfies a non-zero polynomial of degree Then, by Remark 1, will also satisfy a non-zero multi-linear polynomial
for some and Renaming the variables, if necessary, we may assume that Then for some Therefore for all Thus which is a contradiction. So does not satisfy any non-zero polynomial of degree
Theorem. Let be an algebra. Suppose is left primitive, a faithful simple left module and If satisfies a polynomial of degree then and
Proof. Suppose Then there exists some such that either or is a homomorphic image of some subring of In either case, by remark 1, and hence satisfies . Thus by remark 2: , which is nonsense.