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

Abstract

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