  The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is
explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a
consequence, it is proved that in a one-dimensional almost Gorenstein but
non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal.
It is also studied when Ulrich ideals have the same minimal number of
generators.
