  In the present paper we investigate reflexive modules over the endomorphism
algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local
ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka,
and T. T. Phuong and T. Kobayashi concerning the endomorphism algebra of its
maximal ideal. We also explore the question of when the category of reflexive
modules is of finite type, i.e., the base ring has only finitely many
isomorphism classes of indecomposable reflexive modules. We show that, if the
category is of finite type, the ring is analytically unramified and has only
finitely many Ulrich ideals. As a consequence, there are only finitely many
Ulrich ideals are contained in Arf local rings once the normalization is a
local ring.
