Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings

We study finitely generated modules of minimal multiplicity, a notion introduced by Puthenpurakal that extends the classical concept of minimal multiplicity from rings to modules. Our main result characterizes when trace ideals or reflexive ideals yield modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings. As a consequence, we show that a one-dimensional non-Gorenstein reduced local ring with a canonical module has minimal multiplicity if and only if its canonical module has minimal multiplicity as a module. We also construct several examples and compare them with Burch and Ulrich modules, highlighting cases where minimal multiplicity coincides with the Burch or Ulrich property.