Skew polynomial rings are an important source of examples. These are rings of polynomials over rings A with identity in which the indeterminate does not commute with the elements of A, but the multiplication is controlled by an endomorphism and/or a derivation (of some kind) of A. The generalized monoid rings A < G, σ > associated with D-structures σ [3] are analogous to skew polynomials rings, the multiplication being controlled by a family of self-maps of A labelled by the elements of G. They include as special cases the skew monoid rings in the sense of [1]. Amitsur has shown that for every ring A, the Jacobson radical J(A[X]) of the (standard) polynomial ring has the form I[X], where I is a nil ideal of A. Much work has been done on analogues of the Amitsur theorem for various kinds of skew polynomial rings. For a survey see the introduction to [4]. We note also that the question ”if R is nil, must R[X] be quasiregular (i.e. a Jacobson radical ring)?” is equivalent to the Koethe Problem. In general there is interest in connections between the nil radical of the coefficient ring and the Jacobson radical of its skew polynomial rings. Here we take first steps in the investigation of the corresponding problems for generalized monoid rings defined by D-structures. The direct analogue of Amitsur’s Theorem would be the assertion that J(A < G, σ >) = I < G, σ > for some nil ideal I of A. However, I < G, σ > is only defined if I is invariant under all maps of σ. On the other hand, we have an example (used for another purpose in [2]) of a ring A, a monoid G and a D-structure σ, such that the nil radical N(A) of A is non-zero, but the ideal of A < G, σ > generated by N(A) contains an idempotent and so is far from quasiregular. In this case N(A) is not invariant. This investigation is continuing.