On the length equalities for one-dimensional rings

In this article we characterize noetherian local one-dimensional analytically irreducible and residually rational domains image which are non-Gorenstein, the non-negative integer image is equal to τR-1 and image, where τR is the Cohen–Macaulay type of R, image is the conductor of R in the integral closure image of R in its quotient field Q(R) and xR is a minimal reduction of image by giving some conditions on the numerical semi-group v(R) of R.