Density of Polynomial Maps

Abstract

AbstractLet R be a dense subring of End(DV), where V is a left vector space over a division ring D. If dimDV = ∞, then the range of any nonzero polynomial ƒ (X1, … , Xm) on R is dense in End(DV). As an application, let R be a prime ring without nonzero nil one-sided ideals and 0 ≠ a ∈ R. If a f (x1, … , xm)n(xi) = 0 for all x1, … , xm ∈ R, where n(xi ) is a positive integer depending on x1, … , xm, then ƒ (X1, … , Xm) is a polynomial identity of R unless R is a finite matrix ring over a finite field.