The set of trace ideals of curve singularities

We investigate a problem of when commutative local domains have a finite number of trace ideals. The problem is left for the case of dimension one. In this paper, with a necessary assumption, we give a complete answer by using integrally closed ideals. We also explore properties of such domains related to birational extensions, reflexive ideals, and reflexive Ulrich modules. Special attention is given in the case of numerical semigroup rings of non-gap four. We then obtain a criterion for a ring to have a finite number of reflexive ideals up to isomorphism. Non-domains arising from fiber products are also explored.