A classification of one-dimensional local domains based on the invariant $(c-δ)r-δ$
Let $R$ be a one-dimensional, local, Noetherian domain, $\R$ the integral closure of $R$ in its quotient field and $v(R)$ the value set defined by the usual valuation. The aim of the paper is to study the non-negative invariant $b:=(c-δ)r- δ$, where $c, δ, r$ denote the conductor, the length of $\R/R$ and the Cohen Macaulay type, respectively. In particular, the classification of the semigroups $v(R)$ for rings having $b\leq 2(r-1)$ is realized. This method of classification might be successfully utilized with similar arguments but more boring computations in the cases $b\leq q(r-1), $ for reasonably low values of $q$. The main tools are type sequences and the invariant $k$ which estimates the number of elements in $v(R)$ belonging to the interval $[c-e,c), e$ being the multiplicity of $R$.