Let R be an associative ring and Y' a set of polynomials (each in some finite number of noncommuting indeterminates) with integer coefficients. R will be called an IF-ring if and only if for every finite subset S of R there is anf in F which vanishes identically on S. Thus certain kinds of rings can be characterized as 3Frings for appropriate choices of Y: nil rings, with F the set of polynomials xn for all positive integers n; locally nilpotent rings, with 3 the set of monomials x1x2 ... xn in n indeterminates for all n; nilpotent rings, with s the single monomial x1x2... xn for some n; rings satisfying a polynomial identity, with Y finite (or equivalently, with Y containing a single polynomial); and so forth. In addition to Y-rings we shall consider rings satisfying a somewhat stronger property. R is called an F+-ring if and only if for every finitely generated additive subgroup S of R there is anf in F which vanishes identically on S. Clearly every S+-ring is an Y-ring. Also all subrings and homomorphic images of F-rings are F-rings, and similarly for F + -rings. This paper is concerned with determining conditions on , which will ensure that every Y-ring (or F+-ring) has nil commutator ideal (the ideal generated by all commutators [a, b] = ab ba). This intent is similar to that of Drazin in [5] insofar as both are concerned with conditions under which, in a given class of rings, the fact that the commutator ideal is contained in the Jacobson radical implies that the commutator ideal is nil. Drazin's work is self-contained (depending only on Zorn's lemma) but requires that every ring in the class be an f-ring (cf. [5]), whereas here, rather than imposing conditions on our rings individually, we instead confine our attention to classes defined in special ways, i.e., as complete classes of Yor F+-rings, and make use of some relatively special properties of the Jacobson radical (whereas Drazin's arguments would apply equally well to any radical not containing nonzero idempotents). The commutator ideal of R is denoted C(R) and the Jacobson radical J(R). The ring of n x n matrices over R is denoted Rn. R is said to be of characteristic 0, written char (R) =0, if and only if mxA 0 for every nonzero integer m and every nonzero x in R. A linear polynomial means a polynomial which is linear in each of its indeterminates; F will be called linear if and only if each member of F is linear. We consider only sets Y of polynomials each of which is homogeneous in each of its indeterminates. One final assumption about F: Note that if all coeffi-