Summary: 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. If \(A\) has infinite residue field then it follows from a result of Rees that the product of two \(p_g\) ideals is \(p_g\). When \(A\) contains an algebraically closed field \(k\cong A/\mathfrak{m}\) then Okuma, Watanabe and Yoshida [T. Okuma et al., Manuscr. Math. 150, No. 3–4, 499–520 (2016; Zbl 1354.13011)] 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.