On the biggest maximally generated ideal as the conductor in the blowing up ring (Q1911187)

Let \((R, M)\) be a one-dimensional local Cohen-Macaulay ring with total quotient ring \(Q\). A result of \textit{C. Gottlieb} [Commun. Algebra 21, No. 2, 421-425 (1993; Zbl 0779.13006)] states that if \(I\) is an ideal of \(R\) and \(r \in R - \{0\}\), then \(\mu (I) \leq l(R/r R)\), where \(\mu (\;)\) denotes the minimal number of generators and \(l ( )\) denotes length. If \(\mu (I) = l(R/rR)\), \(I\) is said to be maximally generated. Gottlieb also showed that \(R\) has a maximally generated ideal if and only if \(R\) has a superficial element of degree one for \(M\), and if this happens, then \(R\) has a largest maximally generated ideal \(\widehat I\). The authors show that if \(R\) has maximally generated ideals then these are exactly the \(M\)-primary ideals of \(R\) which are also ideals in the ring \(B(M) = \bigcup_{i \geq 1} (M^i : _Q M^i)\). Thus the conductor \((R:_R B(M))\) is the largest maximally generated ideal \(\widehat I\) of \(R\). Under more restrictions on \(R\) the authors give a more explicit construction of \(\widehat I\). In the case that \(R\) is a numerical semigroup ring \(k[S]\), \(S \subseteq \mathbb{N}\), \(k\) a field, it is shown that \(\widehat I\) is generated by monomials. Some further properties of \(\widehat I\) are then studies in terms of its corresponding semigroup ideal in \(S\).