Annihilators in Polynomial Rings

It is known that if f is a divisor of zero in the polynomial ring R [x], where R is a commutative ring, there exists a non-zero element c of R such that cf=0. Proofs of this theorem have been given by McCoy [3], Forsythe [2], Cohen [1], and Scott [4]. It is clear that the theorem as stated does not immediately generalize to polynomials in more than one indeterminate. Moreover, it has been pointed out in Problem 4419 of this MONTHLY (1950, p. 692 and 1952, p. 336) that the theorem itself is not necessarily true for noncommutative rings. The purpose of this note is to obtain a suitable generalization of this result to the case of an arbitrary polynomial ring in any finite number of indeterminates. Let R be an arbitrary ring and R [x1, . . ., X] the ring of polynomials in the indeterminates xi, * * *, Xk, with coefficients in R. If A is a right ideal in R[x1, * .. ., xk], let us denote by Ar the set of right annihilators of A, that is, the ideal consisting of all elements h of the ring R [x1, * .. , Xk] such that Ah = 0. We shall prove the following theorem.