@article { author = {Estaji, Ali and Mahmoudi Darghadam, Ahmad}, title = {Free ideals and real ideals of the ring of frame maps from $\mathcal P(\mathbb R)$ to a frame}, journal = {Algebraic Structures and Their Applications}, volume = {7}, number = {2}, pages = {93-113}, year = {2020}, publisher = {Yazd University}, issn = {2382-9761}, eissn = {2423-3447}, doi = {10.22034/as.2020.1798}, abstract = {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  $\mathcal F_{{\mathcal P}_{K}}( L)$.}, keywords = {Free ideal,$F_{mathcal P}$-realcompact,Lattice-ordered ring,Real ideal,Real Riesz map,Zero-dimensional frame}, url = {https://as.yazd.ac.ir/article_1798.html}, eprint = {https://as.yazd.ac.ir/article_1798_c4382f8727b44d614f536b322fe32db9.pdf} }