A decomposition of polynomials relative to a quasi-homogeneous polynomial

Let F be a quasi-homogeneous polynomial which is the sum of terms +/-x(i)(pi), with p(i) an integer greater than or equal to 2, 1 less than or equal to i less than or equal to n, n greater than or equal to 2. The polynomials in F over R constitute a subring R[F] of R [x] = R [x(1),...,x(n)]. It is shown that R [x] is the direct sum of two R [F]-modules A and B. A is generated by the monomials x(lambda), 0 less than or equal to lambda(i) less than or equal to p(i) - 2, 1 less than or equal to i less than or equal to a. B is the sum of the D-ij R [x]'s, with D-ij = partial derivative(2) F partial derivative(j) - partial derivative(j) F partial derivative(i), 1 less than or equal to i < j less than or equal to n.