@article {
author = {Estaji, Ali Asghar and Hashemi, Ebrahim and Estaji, Ali Akbar},
title = {On Property (A) and the socle of the $f$-ring $Frm(\mathcal{P}(\mathbb R), L)$},
journal = {Categories and General Algebraic Structures with Applications},
volume = {8},
number = {1},
pages = {61-80},
year = {2018},
publisher = {Shahid Beheshti University},
issn = {2345-5853},
eissn = {2345-5861},
doi = {},
abstract = {For a frame $L$, consider the $f$-ring $ \mathcal{F}_{\mathcal P}L=Frm(\mathcal{P}(\mathbb R), L)$. In this paper, first we show that each minimal ideal of $ \mathcal{F}_{\mathcal P}L$ is a principal ideal generated by $f_a$, where $a$ is an atom of $L$. Then we show that if $L$ is an $\mathcal{F}_{\mathcal P}$-completely regular frame, then the socle of $ \mathcal{F}_{\mathcal P}L$ consists of those $f$ for which $coz (f)$ is a join of finitely many atoms. Also it is shown that not only $ \mathcal{F}_{\mathcal P}L$ has Property (A) but also if $L$ has a finite number of atoms then the residue class ring $ \mathcal{F}_{\mathcal P}L/\mathrm{Soc}( \mathcal{F}_{\mathcal P}L)$ has Property (A).},
keywords = {Minimal ideal,Socle,real-valued functions ring,ring with property $(A)$},
url = {http://cgasa.sbu.ac.ir/article_49786.html},
eprint = {http://cgasa.sbu.ac.ir/article_49786_0a546042fb7220c95d9b4ec558b5f554.pdf}
}