On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)
In this article we consider some relations between the topological properties of the spaces X and Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of Min(Cc (X)) is equivalent to the von-Neumann regularity of qc (X), the classical ring of quotients of Cc (X). Furth...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Shahid Beheshti University
2022-07-01
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| Series: | Categories and General Algebraic Structures with Applications |
| Subjects: | |
| Online Access: | https://cgasa.sbu.ac.ir/article_102622_fabfade2e239fe905af15ccfebc0a21e.pdf |
| Summary: | In this article we consider some relations between the topological properties of the spaces X and Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of Min(Cc (X)) is equivalent to the von-Neumann regularity of qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶(𝑋) is a minimal prime ideal of Cc(X) and in this case 𝑀𝑖𝑛(𝐶(𝑋)) and Min(Cc (X)) are homeomorphic spaces. We also observe that if 𝑋 is an Fc-space, then Min(Cc (X)) is compact if and only if 𝑋 is countably basically disconnected if and only if Min(Cc(X)) is homeomorphic with β0X. Finally, by introducing zoc-ideals, countably cozero complemented spaces, we obtain some conditions on X for which Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected. |
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| ISSN: | 2345-5853 2345-5861 |