When ${\mathbb{R}}^X$ is a multiplication $C(X)$-module?

Question

Let $X$ be a compact P-space (i.e., any prime ideal of the ring of continuous functions $C(X)$ on $X$ is a maximal ideal of $C(X)$.). Is ${\mathbb{R}}^X$ a multiplication $C(X)$-module? Recall that an $R$-module $M$ is an $R$-module if any submodule of $M$ has the form $IM$ for some ideal $I$ of $R$.