  The class $M^p$ $(1<p<\infty)$ consists of all holomorphic functions $f$ on
the open unit disk $\Bbb D$ for which $$
\int_0^{2\pi}\left(\log^+Mf(\theta)\right)^p\,\frac{d\theta}{2\pi}<\infty, $$
where $Mf(\theta)=\sup_{0\leqslant r<1}\big\lvert f\big(re^{i\theta}\big)\big
\rvert$. The class $M^p$ equipped with the topology given by the metric
$\rho_p$ defined by $$
\rho_p(f,g)=\left(\int_0^{2\pi}\log^p(1+M(f-g)(\theta))\,
\frac{d\theta}{2\pi}\right)^{1/p}\quad (f,g\in M^p) $$ becomes an $F$-algebra.
In this paper, we consider the ideal structure of the classes $M^p$
$(1<p<\infty)$. Our main result gives a complete characterization of prime
ideals in $M^p$ which are not dense subsets of $M^p$. As a consequence, we
obtaiin a related Mochizuki's result concerning the Privalov classes $N^p$
$(1<p<\infty)$.
