On strong Mori domains
This paper is the continuation of the authors' earlier paper [\textit{F. Wang} and \textit{R. L. McCasland}, Commun. Algebra 25, No. 4, 1285-1306 (1997; Zbl 0895.13010)]. They introduce the notion of strong Mori domains as follows: Let \(R\) denote an integral domain and \(K\) denote its field of quotients. An ideal \(J\) of \(R\) is called a Glaz-Vasconcelos ideal, denoted by \(J\in GV(R)\), if \(J\) is finitely generated and \(J^{-1}= R\) [this notion was introduced by \textit{S. Glaz} and \textit{W. V. Vasconcelos} in Manuscr. Math. 22, 325-341 (1977; Zbl 0367.13002) under the name semi-divisorial ideal)]. Let \(M\) be a torsion-free \(R\)-module. \(M\) is called a \(w\)-module if \(J\in GV(R)\), \(x\in M\otimes K\), and \(Jx\subseteq M\) imply \(x\in M\). \(M\) is a \(w\)-ideal if \(M\) is an ideal of \(R\) and is also a \(w\)-module. A strong Mori module is a \(w\)-module which verifies the ascending chain condition on \(w\)-submodules, and a strong Mori domain (denoted SM domain) is a domain which is a strong Mori module over itself. Strong Mori domains form a proper subclass of Mori domains. In the paper under review the authors show that several Noetherian-like properties hold for SM domains such as the principal ideal theorem, the Hilbert basis theorem and also some ``Krull intersection properties''. In particular, if \(w\operatorname {Max}(R)\) denotes the set of ideals of \(R\) which are maximal in the set of all \(w\)-ideals, the authors show that: \(R\) is an SM domain if and only if \(R_P\) is Noetherian for every \(P\in w\operatorname {Max} (R)\) and each non-zero element of \(R\) lies in only finitely many maximal \(w\)-ideals. Furthermore, if \(R\) is an SM domain, then \(R=\bigcap_{P\in w\operatorname {Max}(R)} R_P\), and if \(\dim(R)=1\) then \(R\) is Noetherian. The authors also discuss \(w\)-multiplication domains and provide some new characterizations of Krull domains (theorem 2.8), such as: \(R\) is a Krull domain if and only if \(R\) is an SM domain and each of its localizations at a maximal \(w\)-ideal is a DVR, or equivalently that \(R\) is an integrally closed SM domain. Finally the authors define the \(w\)-dimension of a domain (namely the supremum of the heights of maximal \(w\)-ideals), and show that the complete integral closure of an SM domain is a Krull domain.
scientific article; zbMATH DE number 1308304
scientific article