Contracted, \(\mathfrak m\)-full and related classes of ideals in local rings (Q2845581)

In a previous paper [\textit{J. Hong, H. Lee, S. Noh} and \textit{D. Rush}, Commun. Algebra 37, No. 8, 2627--2639 (2009; Zbl 1171.13012)], the author and his coauthors compared the following types of ideals: integrally closed, contracted, \(\mathfrak{m}\)-full, full, those satisfying the Rees Property and basically full ideals. Here the author defines some related notions with respect to a regular ideal \(L\) and proves the striking result that in a local domain with infinite residue field a regular ideal \(I\) is \(L\)-contracted for every regular ideal \(L\) if and only if \(I\) is integrally closed. He is also able to prove a similar chain of relationships as in the above mentioned paper for these; in particular, integrally closed \(\Rightarrow\) \(L\)-contracted \(\Rightarrow\) \(L\)-full \(\Rightarrow\) \(L\) Rees Property \(\Rightarrow\) \(L\)-basically full and \(L\)-full \(\Rightarrow\) full for \(L\).