  Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two. We
define an $\mathfrak{m}$-primary ideal $I$ to be a $p_g$-ideal if the Rees
algebra $A[It]$ is a Cohen-Macaulay normal domain. When $A$ contains an
algebraically closed field $k \cong A/\mathfrak{m}$ then Okuma, Watanabe and
Yoshida proved that $A$ has $p_g$-ideals and furthermore product of two
$p_g$-ideals is a $p_g$ ideal. In this article we show that if $A$ is an
excellent normal domain of dimension two containing a field $k \cong
A/\mathfrak{m}$ of characteristic zero then also $A$ has $p_g$-ideals.
Furthermore product of two $p_g$-ideals is $p_g$.
