Counterexamples for certain star operations on ideals (Q2367208)

For an integral domain \(R\) the \(v\)-operation is defined on the set of nonzero fractional ideals of \(R\) by \(I_ v= \bigl[ R:[R:I] \bigr]=\bigcap \{Rx:Rx \supseteq I \}\) and the \(t\)-operation is defined by \(I_ t=\bigcup\{J_ v:0 \neq J \subseteq I\) is finitely generated\}. For \(n \geq 2\), \textit{J. R. Hedstrom} and \textit{E. G. Houston} [J. Pure Appl. Algebra 18, 37-44 (1980; Zbl 0462.13003)] defined the star operations \(t_ n\) on \(R\) by \(I_{t_ n}=\bigcup \bigl\{ (a_ 1,\ldots,a_ n)_ v:\text{ each } a_ i \in I \bigr\}\). Also, set \(I_{t_ 1}=I\). They asked whether the \(t\)-operation and the \(t_ 2\)- operation were the same. -- This paper shows that this is not the case. It is shown that for each \(1 \leq r<m\), there is a Noetherian domain \(R\) and an ideal \(I\) of \(R\) so that \(I_{t_ r} \subset I_{t_{r+1}} \subset \cdots \subset I_{t_ m}\) such that \(I_{t_ m}\) is a maximal divisorial ideal of \(R\). Here \(R\) can be chosen to be a graded algebra over a field or a local domain. It is also shown that for each natural number \(r\) there is a domain \(R\) and an ideal \(I\) of \(R\) so that \(I=I_{t_ r} \subset I_{t_{r+1}} \subset \cdots\) and such that \(I_ t\) is a maximal divisorial ideal of \(R\). Moreover, \(R\) can be chosen to be a graded algebra over a field or a quasi-local domain.