On \({\mathfrak m}\)-full powers of parameter ideals (Q2372711)

Let \((A,\mathfrak m)\) be a local ring of dimension \(d\). Let \(I\) be an ideal of \(A\). Then \(I\) is said to be \(\mathfrak m\)-full if \(\mathfrak m I\colon x=I\), for some \(x\in \mathfrak m\). This notion was introduced by Rees and studied by other authors, like \textit{S. Goto} [J. Algebra 108, 151--160 (1987; Zbl 0629.13004)] and \textit{M. Morales, Ngô Viêt Trung} and \textit{O. Villamayor} [J. Algebra 129, No.~1, 96--102 (1990; Zbl 0701.13004)], regarding the structure of integrally closed ideals. The main result of the article under review is that, if \(Q\) is a parameter ideal of a noetherian local ring \((A,\mathfrak m)\) such that depth\(A>0\) and \(Q^n\) is \(\mathfrak m\)-full for some integer \(n\geq 1\), then \(A\) is regular and \(\mathfrak m/Q\) is cyclic. As a corollary it follows that, if \((A,\mathfrak m)\) is a noetherian local ring and \(Q\) is a parameter ideal, then the following conditions are equivalent: (1) \(A\) is a regular local ring and \(\mathfrak m/Q\) is cyclic; (2) \(Q\) is integrally closed in \(A\); (3) \(Q^n\) is integrally closed in \(A\), for some \(n\geq 1\); (4) \(Q^n\) is integrally closed in \(A\), for all \(n\geq 1\). The contribution of the author to the proof of this corollary is implication \((3)\Rightarrow (2)\). The equivalence of (1) and (2) is due to \textit{S. Goto} [loc. cit.]. The proof of the main result is preceded by preliminary results on Ratliff-Rush closures, generalized Cohen-Macaulay rings and \(\mathfrak m\)-full ideals.