Noetherian properties in monoid rings

Let R be a commutative unitary ring and let M be a commutative monoid. The monoid ring R[M] is considered as an M-graded ring where the homogeneous elements of degree s are the elements of the form aXs, a R, s M. If each homogeneous ideal of R[M] is finitely generated, we say R[M] is gr-Noetherian. We denote the set of homogeneous prime ideals of R[M] by h-Spec(R[M]). Results are given which illuminate the difference between the Noetherian and gr-Noetherian conditions on a monoid ring, and also the difference between Spec(R[M]) being Noetherian and h-Spec(R[M]) being Noetherian. Applications include a variation of the Mori–Nagata theorem and some results on group rings which are ZD-rings, Laskerian rings or N-rings.