Derivations and right ideals of algebras

Let R be a K-algebra acting densely on VD, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X1,…,Xt) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X1,…,Xt). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x1,…,xt))⩽m for all x1,…,xt∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(VD) of finite rank. In addition, if f(X1,…,Xt) is multilinear, then b can be chosen such that rank(b)⩽2(6t+13)m+2.