Noetherian properties in monoid rings

✍ Scribed by David E. Rush

• Book ID: 104152684
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 255 KB
• Volume: 185
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(03)00103-8

✦ Synopsis

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 ÿnitely 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 di erence between the Noetherian and gr-Noetherian conditions on a monoid ring, and also the di erence 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.