Noetherian rings graded by an abelian group
Let \(G\) be an abelian group. \(G\)-graded rings and \(G\)-graded modules are defined in an obvious way. In the case of \(G = \mathbb{Z}^n\), some homological properties of a \(G\)-graded ring are determined by their local data at graded prime ideals. But this is not true anymore for an arbitrary abelian group \(G\). The author studies \(G\)-graded rings and \(G\)- graded modules for an arbitrary abelian group \(G\). Section one consists of preliminaries. In section two, among others, the author defines the \(G\)-Bass number and establishes a relation between the \(G\)-Bass number and the ordinary Bass number in the case where a ring is noetherian. Using this relation, it is shown that if \(R\) is a noetherian \(G\)-graded ring, the following are equivalent for a \(G\)-prime ideal \({\mathfrak p}\): (1) \(R_{({\mathfrak p})}\) is Cohen-Macaulay (resp. Gorenstein). (2) \(R_P\) is Cohen-Macaulay (resp. Gorenstein) for some \(P \in \text{Ass}_R (R/{\mathfrak p})\). (3) There exists \(P \in \text{Spec} (R)\) such that \(P^* = {\mathfrak p}\) and \(R_P\) is Cohen-Macaulay (resp. Gorenstein). \((R_{({\mathfrak p})}\) is the localization with respect to the multiplicative closed set of all homogeneous elements not in \({\mathfrak p}\). \(P^*\) is the largest graded ideal contained in \(P\).) In section three the \(G\)-canonical module is defined and studied. -- In the final section a condition for a \(G\)-prime ideal to be really a prime ideal is given.
scientific article; zbMATH DE number 821707