In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called **strongly** $z$-**ideal** if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} \mathcal{M_{\alpha}}$$ Where each $\mathcal{M_{\alpha}}$ is a maximal ideal of $C(X)$.

Question:If $I$ and $J$ are two strongly $z$-ideals of $C(X)$, Is the ideal $I+J$ strongly $z$-ideal or all of $C(X)$?

PS: If $X$ is a $P$- space, then each ideal of $C(X)$ is strongly $z$- ideal, And conversly if each ideal of $C(X)$ is strongly $z$- ideal than $X$ is a $P$-space.