On the length equalities for one-dimensional rings

In this article we characterize noetherian local one-dimensional analytically irreducible and residually rational domains (R,mR)(R,mR) which are non-Gorenstein, the non-negative integer ℓ*(R)=τR·ℓ(R/C)-ℓ(R¯/R) is equal to τR-1τR-1 and ℓ(R/(C+xR))=2ℓ(R/(C+xR))=2, where τRτR is the Cohen–Macaulay type of R  , CC is the conductor of R   in the integral closure R¯ of R   in its quotient field Q(R)Q(R) and xR   is a minimal reduction of mm by giving some conditions on the numerical semi-group v(R)v(R) of R.