Construction of ideal systems with nice Noetherian properties (Q2752911)

Let \(D\) be an integral domain with quotient field \(K\). For a non-zero fractional ideal \(I\) of \(D\), \(I_v=(I^{-1})^{-1}\) and \(I_t=\bigcup \{J_v \mid J\subseteq I\) is a nonzero finitely generated ideal\}. \textit{F. Wang} and \textit{R. L. McCasland} [Commun. Algebra 25, 1285-1306 (1997; Zbl 0895.13010)] introduced the \(w\)-operation on modules. In the case of a non-zero fractional ideal \(I\), they defined \(I_w=\{x\in K\mid Jx\subseteq I\) for some finitely generated ideal \(J\) of \(D\) with \(J^{-1}=D\}\). Later, the reviewer and S. Cook [\textit{D. D. Anderson} and \textit{S. J. Cook}, Commun. Algebra 28, 2461-2475 (2000; Zbl 1043.13001)] generalized the \(w\)-operation construction to an arbitrary star-operation * as follows. For a non-zero fractional ideal \(I\), define the new star-operation \(*_w\) by \(I^{*_w}= \{x\in K\mid Jx\subseteq I\) for some finitely generated ideal \(J\) of \(D\) with \(J^*=D\}\); so for the star-operation \(t\), we have \(w=t_w\). The star-operation \(*_w\) has nicer properties than the original star-operation *. For example, \(I^*=\bigcap \{I_P\mid P\) is a maximal \(*_w\)-ideal\}. In the paper under review the \(*_w\) construction is generalized to a general \(x\)-system in the sense of Aubert.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00012].