Note on spectral semistar operations. II (Q2858128)

Let \(R\) be an integral domain, \(K\) its quotient field, \(\bar{F}(R)\) the set of all non-zero \(R\)-submodules of \(K\) and \(\mathrm{Spec}(R)\) the set of all prime ideals of \(R\). For any subset \(\Delta\) of \(\mathrm{Spec}(R)\), the mapping \(*_{\Delta}: E\mapsto E^{*_{\Delta}}=\displaystyle\bigcap_{P\in \Delta}ER_{P}\) for every \(E\in \bar{F}(R)\) defines a semistar operation on \(R\). A semistar operation \(*\) on \(R\) is said to be spectral if \(*=*_{\Delta}\) for some \(\Delta\subseteq \mathrm{Spec}(R)\). In this note the author proved the existence of an infinite dimensional non-local Prüfer domain \(R\) which has infinitely many non-spectral semistar operations \(*\) with \(R^{*}=R\).