On structure of domains with quadratic growth

Abstract

Let F be a finite field, and let R be an affine F-algebra which is a domain of Gelfand–Kirillov dimension smaller than 3. Let m , n be natural numbers. Assume that x ∈ R is transcendental over F and y 1 , … , y n ∈ R are such that ∑ i , j ⩽ m α i , j x i y k x j = 0 , for some α i , j ∈ F (not all equal to 0) and each k ⩽ n . It is shown that either R satisfies a polynomial identity or else the subalgebra of R generated by y 1 , y 2 , … , y n and x has Gelfand–Kirillov dimension 1. From this we deduce that a finitely generated domain over F with quadratic growth and with an infinite centre satisfies a polynomial identity (is a PI domain). Moreover, the centralizer of a non-algebraic element in a finitely generated domain with quadratic growth over finite field is a PI domain.