It is shown that the annihilator of each finitely generated ideal of R[3XA C\EA]I where R is a commutative von Neumann regular ring with identity, is principal; this generalizes a recent result of P. J. McCarthy. In a recent paper [5], P. J. McCarthy showed that the ring of polynomials in one indeterminate over a commutative von Neumann regular ring R with identity is semihereditary. 1 Based upon results of W. Vasconcelos [8] and C. U. Jensen [3], McCarthy's proof rested upon the establishment of the following result, which we label as Theorem A. Theorem A. If f(X) = aO + a1X + *-+ anX' is in R[X] and if ei is an idempotent generator of the ideal aiR for each i, then the annihilator of f(X) is the principal ideal of R[X] generated by (1eo)(1 el) *.. (1 en). In this note we prove a result (Theorem C) that includes Theorem A; we begin with more general considerations. Let S be a commutative ring, let XXA } A be a set of indeterminates over S, and let f be an element of the polynomial ring S[IXA}]. If Af is the ideal of S generated by the coefficients of f, then it is clear that each element of B[UXA}], where B is the annihilator of the ideal A/' annihilates f. In the next result we give sufficient conditions in order that B[UX ,j] should be the annihilator of f. Proposition B.2 In order that B[XXAfl should be the annihilator of f, it is sufficient that either of the following conditions is satisfied. (1) The ideal Af is idempotent. (2) The ring S contains no nonzero nilpotent element. Proof. Assume that (1) is satisfied. Since Af is finitely generated, it Received by the editors February 25, 1974. AMS (MOS) subject classifications (1970). Primary 13B25, 13F20; Secondary 13C99. 1A statement of this result is also contained in [4]. 2A form of this result, for power series rings, is contained in [1]; the paper [7], by J. Ohm and D. Rush, considers problems related to Proposition B. Copyright ? 1975, Anmerican Mathematical Society