Density of Polynomial Maps

Let R be a dense subring of End(V-D), where V is a left vector space over a division ring D. If dim V-D = infinity, then the range of any nonzero polynomial f(X-1, ..., X-m) on R is dense in End(V-D). As an application, let R be a prime ring without nonzero nil one-sided ideals and 0 not equal a is an element of R. If a f(x(1), ..., x(m))(n(xi)) = 0 for all x(1), ..., x(m) is an element of R, where n(x(i)) is a positive integer depending on x(1), ..., x(m) then f(X-1, ..., X-m) is a polynomial identity of R unless R is a finite matrix ring over a finite field.