Let $p$ be an odd prime number and $F$ a number field. Let $K=F\left({\zeta }_p\right)$ and $\mathrm{\Delta }=\mathrm{G}\mathrm{a}\mathrm{l}(K/F)$. Let ${𝒮}_{\mathrm{\Delta }}$ be the Stickelberger ideal of the group ring $\mathbf{Z}\left[\mathrm{\Delta }\right]$ 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 ${𝒮}_{\mathrm{\Delta }}$ annihilates the $p$-ideal class group of $K$. In this paper, we study some properties of the ideal ${𝒮}_{\mathrm{\Delta }}$,and check whether or not a subfield of $\mathbf{Q}\left({\zeta }_p\right)$ satisfies the above property.