\(w\)-overrings of \(w\)-Noetherian rings (Q2915436)

Let \(R\) be a commutative ring with identity. An overring of \(R\) is a ring between \(R\) and its total quotient ring \(T(R)\). If \(R\) is a one-dimensional Noetherian domain, then every overring of \(R\) is Noetherian by the Krull-Akizuki Theorem, see e.g. Theorem 93 in \textit{I. Kaplansky} [Commutative rings. 2nd revised ed. Chicago-London: The University of Chicago Press. (1974; Zbl 0296.13001)]. This theorem was generalized in \textit{J. R. Matijevic} [Proc. Am. Math. Soc. 54, 49--52 (1976; Zbl 0318.13018)] for arbitrary Noetherian rings by using the idea of the global transform of \(R\) defined as NEWLINE\[NEWLINER^g=\{x\in T(R)\mid M_1M_2\cdots M_nx \subseteq R \text{ for some } M_i\in\max(R)\},NEWLINE\]NEWLINE where \(\max(R)\) denotes the set of maximal ideals of \(R\).NEWLINENEWLINE In the paper under review an \(w\)-analogue of Matijevic's result is proved. By using the concept of Glaz-Vasconcelos ideals of \(R\) one can define the set of \(w\)-ideals of \(R\) as in \textit{H. Yin, F. Wang, X. Zhu and Y. Chen} [J. Korean Math. Soc. 48, 207--222 (2011; Zbl 1206.13005)]. Let the \(w\)-global transform of \(R\) be the set NEWLINE\[NEWLINER^{wg}=\{x\in T(R)\mid M_1M_2\cdots M_nx\subseteq R \text{ for some } M_i\in w\text{-}\max(R)\}.NEWLINE\]NEWLINE Now, it is proved that if \(R\) is an \(w\)-Noetherian ring, and \(T\) is an \(w\)-overring of \(R\) with \(T\subseteq R^{wg}\), then \(T\) has ACC on regular \(w\)-ideals. If, moreover, \(w-\dim(R)\leq 1\), then also \(R^{wg}=T(R)\) and \(w -\dim(T)\leq 1\) hold.