  The notion of $p_g$-ideals for normal surface singularities has been proved
to be very useful. On the other hand, the core of ideals has been proved to be
very important concept and also very mysterious one. However, the computation
of the core of an ideal seems to be given only for very special cases. In this
paper, we will give an explicit description of the core of $p_g$-ideals of
normal surface singularities. As a consequence, we give a characterization of
rational singularities using the inclusion of the core of integrally closed
ideals.
