TY - JOUR
ID - 49786
TI - On Property (A) and the socle of the $f$-ring $Frm(mathcal{P}(mathbb R), L)$
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Estaji, Ali Asghar
AU - Hashemi, Ebrahim
AU - Estaji, Ali Akbar
AD - Department of Mathematics, Shahrood University of Technology, Shahrood, Iran.
AD - Department of Mathematics, Shahrood University of Technology
AD - Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Y1 - 2018
PY - 2018
VL - 8
IS - 1
SP - 61
EP - 80
KW - Minimal ideal
KW - Socle
KW - real-valued functions ring
KW - ring with property $(A)$
DO -
N2 - 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).
UR - http://cgasa.sbu.ac.ir/article_49786.html
L1 - http://cgasa.sbu.ac.ir/article_49786_0a546042fb7220c95d9b4ec558b5f554.pdf
ER -