  In this paper we consider reduced (non-normal) commutative noetherian rings
$R$. With the help of conductor ideals and trace ideals of certain $R$-modules
we deduce a criterion for a reflexive $R$-module to be closed under
multiplication with scalars in an integral extension of $R$. Using results of
Greuel and Kn\"orrer this yields a characterization of plane curves of finite
Cohen--Macaulay type in terms of trace ideals.
  Further, we study one-dimensional local rings $(R,\mathfrak{m})$ such that
that their normalization is isomorphic to the endomorphism ring
$\mathrm{End}_R(\mathfrak{m})$: we give a criterion for this property in terms
of the conductor ideal, and show that these rings are nearly Gorenstein.
Moreover, using Grauert--Remmert normalization chains, we show the existence of
noncommutative resolutions of singularities of low global dimensions for curve
singularities.
