Noetherian domains in which every ideal is isomorphic to a trace ideal

This paper seeks an answer to the following question: Let R be a Noetherian ring with depth(R) ≤ 1. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain R with depth(R) ≤ 1, every ideal is isomorphic to a trace ideal if and only if either R is a DVR or R is one-dimensional divisorial domain, M is a principal ideal of M-1 and M-1 posses the property that every ideal of M-1 is isomorphic to a trace ideal of M-1. Next, we globalize our result by showing that a Noetherian domain R with depth(R) ≤ 1 has every ideal isomorphic to a trace ideal if and only if either R is a PID or R is one-dimensional divisorial domain, every invertible ideal of R is principal and for every non-invertible maximal ideal M of R, M is a principal ideal of M-1 and every ideal of M-1 is isomorphic to a trace ideal of M-1. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.