Free ideals and real ideals of the ring of frame maps from $mathcal P(mathbb R)$ to a frame

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 $fin mathcal F_{mathcal P}( L)$ such that  ${uparrow}f(-frac 1n, frac 1n)$ is compact for any $ninmathbb N$ and the subring  $mathcal F_{{mathcal P}_{K}}( L)$ is the family of all $fin mathcal F_{mathcal P}( L)$ 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  $mathcal F_{{mathcal P}_{K}}( L)$.