Remarks on P. I. algebras over finite fields (Q1183298)

Let \(F=F_ 1\subset F_ 2\subset\dots\) be a chain of finite fields. There exists a chain of discrete valuation rings \(R=R_ 1\subset R_ 2\subset\dots\) with maximal ideals \(P=P_ 1\subset P_ 2\subset\dots\) such that \(F_ n=R_ n/P_ n\) and \(P_ n=P_{n+1}\cap R_ n\). The author considers the polynomial identities and different codimensions for an \(F\)-algebra \(A=B/PB\) where \(B\) is an \(R\)-torsion free PI-algebra. Forming \(A_ n=F_ n\otimes A\) and \(B_ n=R_ n\otimes B\) the limits for the codimensions are considered. In this way the study of PI-algebras over finite fields is related to that of PI-algebras over the rationals. As an application of the method the polynomial identities for matrix algebras over finite fields are studied.