  The authors introduced the notion of $p_g$-ideals for two-dimensional
excellent normal local domain over an algebraicaly closed field in terms of
resolution of singularities. In this note, we give several ring-theoretic
characterization of $p_g$-ideals. For instance, an $m$-primary ideal $I \subset
A$ is a $p_g$-ideal if and only if the Rees alegbra $\mathcal{R}(I)$ is a
Cohen-Macaulay normal domain.
