The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small possible. In this paper we call such far-flung Gorenstein rings. investigate in relation with endomorphism algebras maximal ideals and numerical semigroup show that solution Rohrbach problem additive number theory provides an upper bound for multiplicity Reflexive modules over are also studied.