Noetherian properties in monoid rings

Abstract

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 aX s , 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.