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Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1‐dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.