Let $p$ be an odd prime number and $F$ a number field. Let $K=F(\zeta_p)$ and $\Delta=\mathrm{Gal}(K/F)$ . Let $\mathscr{S}_{\Delta}$ be the Stickelberger ideal of the group ring $\mathbf{Z}[\Delta]$ defined in the previous paper [8]. As a consequence of a $p$ -integer version of a theorem of McCulloh [15], [16], it follows that $F$ has the Hilbert-Speiser type property for the rings of $p$ -integers of elementary abelian extensions over $F$ of exponent $p$ if and only if the ideal $\mathscr{S}_{\Delta}$ annihilates the $p$ -ideal class group of $K$ . In this paper, we study some properties of the ideal $\mathscr{S}_{\Delta}$ ,and check whether or not a subfield of $\mathbf{Q}(\zeta_p)$ satisfies the above property.