Polynomial Identity Algebras

Abstract

Definitions. A ring R is a polynomial identity ring (or PI ring for short) if R satisfies a monic polynomial f ∈ ℤ 〈X〉. Here, ℤ〈X〉 is the free ℤ-algebra on a finite set, X={x ₁,...,x _m } and to say that R satisfies f=f(x ₁,...,x _m ) means f(r ₁,...,r _m )=0 for all r ₁,...,r _m ∈ R. That f is monic means that at least one of the monomials of highest degree in f has coefficient 1; here degree refers to total degree. The minimal degree of a PI ring R is the least degree of a monic polynomial identity for R.