Noetherian properties in monoid rings. (Q1414066)

Let \(R\) be a commutative unitary ring, \(M\) a commutative monoid, and \(S = R[M]\) the monoid ring. Recall that \(S\) is said to be gr-Noetherian if every homogeneous ideal of \(S\) is finitely generated. Moreover, \(\text{Spec}(S)\) is said to be Noetherian if \(S\) satisfies the ascending chain condition (ACC) on radical ideals, and \(\text{h-Spec}(S)\) is said to be Noetherian if \(S\) satisfies the ascending chain condition on homogeneous radical ideals. This paper is directed towards determining when these conditions hold for various classes of monoids \(M.\) For example, it is proved (theorem 2.4) that if \(M\) is cancellative and torsion-free, \(R[M]\) is gr-Noetherian if and only if \(R\) is Noetherian and \(M\) has ACC on ideals. (Thus if \(M\) is a torsion-free group, \(R[M]\) is gr-Noetherian if and only if \(R\) is Noetherian.) By way of contrast, a result of \textit{L. Budach} [Monatsber. Deutsch. Akad. Wiss. Berlin 6, 85--88 (1964; Zbl 0123.01802)] (which applies to any commutative monoid \(M\)) shows that \(R[M]\) is Noetherian if and only if \(R\) is Noetherian and \(M\) is finitely generated. In \(\S\)4 of the paper, Noetherianity of \(\text{Spec}(R[M])\) is determined for those \(M\) with \(M/U(M)\) periodic, where \(U(M)\) denotes the group of units of \(M.\) Various applications of these results are also proved.