Let R be a K-algebra acting densely on V-D, where K is a commutative ring with unity and Visa right vector space over a division K-algebra D. Let rho be a nonzero right ideal of R and let f (X-1, ..., X-t) be a nonzero polynomial over K with constant term 0 such that mu R not equal 0 for some coefficient mu of f (X-1, ..., X-t). Suppose that d : R --> R is a nonzero derivation. It is proved that if rank(d(f(x(1), ..., x(t)))) <= m for all x(1), ..., x(t) is an element of rho and for some positive integer m, then either rho is generated by an idempotent of finite rank or d = ad(b) for some b is an element of End(V-D) of finite rank. In addition, if f (X-1, ..., X-t) is multilinear, then b can be chosen such that rank(b) <= 2(6t + 13)m + 2. (C) 2009 Elsevier Inc. All rights reserved.