Core of an ideal in Prüfer domains (Q6564634)

Let \(R\) be a commutative integral domain, and let \(I\) be a nonzero ideal of \(R\). Recall that the \emph{core} of \(I\) is the intersection of all reductions of \(I\), where a \emph{reduction} of \(I\) is an ideal \(J\) of \(R\) such that \(JI^{n}=I^{n+1}\) for some positive integer \(n\). The ideal \(I\) is \emph{stable} is \(I\) is invertible in its endomorphism ring \((I:I)\). If \(R\) is a Prüfer domain, then \(I^{2}I^{-1}\subseteq \mathrm{core}(I)\), and \(I\) is said to have a \emph{small core} if \(\mathrm{core}(I)=I^{2}I^{-1}\). The domain \(R\) is of \emph{finite character} if each nonzero ideal is contained in just finitely many maximal ideals of \(R\).\N\NHere are some of the authors' results:\N\NIf \(R\) is a Prüfer domain, then the ideal \(I\) is stable if and only if \(I\) has a finitely generated reduction. Also each stable ideal has a small core. Further conditions are presented for an ideal of a Prüfer domain to have a small core. A Prüfer domain has finite character if and only if every ideal has a small core.\N\NIf \(R\) is an arbitrary domain, and \(I\) is flat, or each prime ideal containing \(I\) is contained in a unique maximal ideal of \(R\), then\\\N\(\mathrm{core}(I) =\bigcap_{M\in \mathrm{Max}(R)}\mathrm{core}_{R_{M}}(IR_{M})\). The authors characterize the \(h\)-local property of a domain in terms of the core concept.