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

Abstract

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