  This paper mainly focuses on commutative local domains of dimension one. We
then obtain a criterion for a ring to have a finite number of trace ideals in
terms of integrally closed ideals. We also explore properties of such rings
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.
