Let $\mathcal F_{\mathcal P}( L)$ ($\mathcal F_{\mathcal P}^{*}( L)$) be the $f$-rings of all (bounded) frame maps from $\mathcal P(\mathbb R)$ to a frame $L$. $\mathcal F_{{\mathcal P}_{\infty}}( L)$ is the family of all $f\in \mathcal F_{\mathcal P}( L)$ such that ${\uparrow}f(-\frac 1n, \frac 1n)$ is compact for any $n\in\mathbb N$ and the subring $\mathcal F_{{\mathcal P}_{K}}( L)$ is the family of all $f\in \mathcal F_{\mathcal P}( L)$ such that ${{\,\mathrm{coz}\,}}(f)$ is compact. We introduce and study the concept of real ideals in $\mathcal F_{\mathcal P}( L)$ and $\mathcal F_{\mathcal P}^*( L)$. We show that every maximal ideal of $\mathcal F_{\mathcal P}^{*}( L)$ is real, and also we study the relation between the conditions ``$L$ is compact" and ``every maximal ideal of $\mathcal F_{\mathcal P}(L)$ is real''. We prove that for every nonzero real Riesz map $\varphi \colon \mathcal F_{\mathcal P}( L)\rightarrow \mathbb R$, there is an element $p$ in $\Sigma L$ such that $\varphi=\widetilde {p_{{{\,\mathrm{coz}\,}}}}$ if $L$ is a zero-dimensional frame for which $B(L)$ is a sub-$\sigma$-frame of $L$ and every maximal ideal of $\mathcal F_{\mathcal P}( L)$ is real. We show that $\mathcal F_{{\mathcal P}_{\infty}}(L)$ is equal to the intersection of all free maximal ideals of $ \mathcal F_{\mathcal P}^{*}(L) $ if $B(L)$ is a sub-$\sigma$-frame of a zero-dimensional frame $L$ and also, $\mathcal F_{{\mathcal P}_{K}}(L)$ is equal to the intersection of all free ideals $\mathcal F_{\mathcal P}( L)$ (resp., $\mathcal F_{\mathcal P}^*( L)$) if $L$ is a zero-dimensional frame. Also, we study free ideals and fixed ideals of $\mathcal F_{{\mathcal P}_{\infty}}( L)$ and $\mathcal F_{{\mathcal P}_{K}}( L)$.