On Property (A) and the socle of the $f$-ring $Frm(\mathcal{P}(\mathbb R), L)$ | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 7، دوره 8، شماره 1، فروردین 2018، صفحه 61-80 اصل مقاله (623.33 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.29252/cgasa.8.1.61 | ||
| نویسندگان | ||
| Ali Asghar Estaji1؛ Ebrahim Hashemi2؛ Ali Akbar Estaji* 3 | ||
| 1Department of Mathematics, Shahrood University of Technology, Shahrood, Iran. | ||
| 2Department of Mathematics, Shahrood University of Technology | ||
| 3Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. | ||
| چکیده | ||
| 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). | ||
| کلیدواژهها | ||
| Minimal ideal؛ Socle؛ real-valued functions ring؛ ring with property $(A)$ | ||
| مراجع | ||
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