On the length equalities for one-dimensional rings (Q819792)

Let \(R\) be a noetherian local one-dimensional analytically irreducible and residual rational domain which is not Gorenstein. Let \(\ell^*(R) = \tau_R\ell(R/C)-\ell(\overline R/R)\), where \(\tau_R\) is the Cohen-Macaulay type of \(R\), \(C\) is the conductor of \(R\) in the integral closure \(\overline R\). In this paper, the authors characterize the rings \(R\) with \(\ell^*(R) = \tau_R-1\) and \(\ell(R/C+xR) = 2\), where \(xR\) is a minimal reduction of the maximal ideal, by means of the numerical semigroup of \(R\). Moreover, they also give a characterization of almost Gorenstein semigroup rings \(R\) with \(\ell^*(R) = \tau_R-1\).