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On CP-frames | ||
| Journal of Algebra and Related Topics | ||
| دوره 9، شماره 1، شهریور 2021، صفحه 109-119 اصل مقاله (131.15 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2021.18801.1252 | ||
| نویسندگان | ||
| A. A. Estaji* 1؛ M. Robat Sarpoushi2 | ||
| 1Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran | ||
| 2Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran | ||
| چکیده | ||
| Let $\mathcal{R}_c( L)$ be the pointfree version of $C_c(X)$, the subring of $C(X)$ whose elements have countable image. We shall call a frame $L $ a $CP$-frame if the ring $\mathcal{R}_c( L)$ is regular. % The main aim of this paper is to introduce $CP$-frames, that is $\mathcal{R}_c( L)$ is a regular ring. We give some We give some characterizations of $CP$-frames and we show that $L$ is a $CP$-frame if and only if each prime ideal of $\mathcal{R}_c ( L)$ is an intersection of maximal ideals if and only if every ideal of $\mathcal{R}_c ( L)$ is a $z_c$-ideal. In particular, we prove that any $P$-frame is a $CP$-frame but not conversely, in general. In addition, we study some results about $CP$-frames like the relation between a $CP$-frame $L$ and ideals of closed quotients of $L$. Next, we characterize $CP$-frames as precisely those $L$ for which every prime ideal in the ring $\mathcal{R}_c ( L)$ is a $z_c$-ideal. Finally, we show that this characterization still holds if prime ideals are replaced by essential ideals, radical ideals, convex ideals, or absolutely convex ideals. | ||
| کلیدواژه‌ها | ||
| P-frame؛ CP-frame؛ regular ring؛ z-ideal؛ z-good ring | ||
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آمار تعداد مشاهده مقاله: 648 تعداد دریافت فایل اصل مقاله: 573 |
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