Annihilating properties of ideals generated by coefficients of polynomials and power series

In this paper, we study the annihilating properties of ideals generated by coefficients of polynomials and power series which satisfy a structural equation. We first show that if [Formula: see text] for polynomials [Formula: see text] over any ring [Formula: see text], then for any [Formula: see text], there exist positive integers [Formula: see text] and [Formula: see text] such that [Formula: see text] and [Formula: see text], whenever [Formula: see text] and [Formula: see text]. Next we prove that if [Formula: see text] for power series [Formula: see text] over any ring [Formula: see text], then for any [Formula: see text], there exist positive integers [Formula: see text] and [Formula: see text] such that [Formula: see text] when [Formula: see text] and [Formula: see text], [Formula: see text] for each [Formula: see text].