On socle and Property (A) of the f-ring $Frm (mathcal{P} (mathbb R), L) $

A topoframe, denoted by $L_{ tau}$, is a pair $(L, tau)$ consisting of a frame $L$ and a subframe $ tau $ all of whose elements are complementary elements in$L$. $f$-ring $mathcal{R}(L_{ tau})$ is equal to the set $${fin Frm(mathcal{P}(mathbb R), L): f(mathfrak{O}(mathbb R))subseteq tau} .$$ In this paper, for every complemented element $ain L$ with $a, a'in tau$, we introduce anidempotent element $f_{a}$ belong to $mathcal{R}(L_{ tau})$ and we show that an ideal $I$ of $mathcal{R}(L_{ tau})$ is minimal if and only if there exists an atom $a$ of $L$ such that $I$ is generated by $ f_a$ if and only if there exists an atom $a$ of $L$ such that $ I={fin mathcal{R}(L_{ tau}): coz(f)leq a} $. Also, we prove that the socle of $f$-ring $mathcal{R}(L_{ tau})$ consists of those $f$ for which $coz (f)$ is a join of finitely many atoms and finally, we show that the $f$-ring $mathcal{R}(L_{ tau})$ has Property (A) and if $L$ has a finite number of atoms then the residue class ring $ frac{mathcal{R}(L_{ tau})}{Soc (mathcal{R}(L_{ tau}))}$ has Property (A).