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...
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| Format: | Article |
| Language: | English |
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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 |
| _version_ | 1818497168852385792 |
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| author | Zahra Keshtkar Rostam Mohamadian Mehrdad Namdari Maryam Zeinali |
| author_facet | Zahra Keshtkar Rostam Mohamadian Mehrdad Namdari Maryam Zeinali |
| author_sort | Zahra Keshtkar |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-10T18:42:17Z |
| format | Article |
| id | doaj.art-c36e2e58b3e94432be9e226b431448a0 |
| institution | Directory Open Access Journal |
| issn | 2345-5853 2345-5861 |
| language | English |
| last_indexed | 2024-12-10T18:42:17Z |
| publishDate | 2022-07-01 |
| publisher | Shahid Beheshti University |
| record_format | Article |
| series | Categories and General Algebraic Structures with Applications |
| spelling | doaj.art-c36e2e58b3e94432be9e226b431448a02022-12-22T01:37:37ZengShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532345-58612022-07-011718510010.52547/cgasa.2022.102622102622On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)Zahra Keshtkar0Rostam Mohamadian1Mehrdad Namdari2Maryam Zeinali3Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.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.https://cgasa.sbu.ac.ir/article_102622_fabfade2e239fe905af15ccfebc0a21e.pdfthe space of minimal prime idealsstrongly zero-dimensional spacecountably basically disconnected spacecountably cozero complemented space$z^0_c$-ideals |
| spellingShingle | Zahra Keshtkar Rostam Mohamadian Mehrdad Namdari Maryam Zeinali On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋) Categories and General Algebraic Structures with Applications the space of minimal prime ideals strongly zero-dimensional space countably basically disconnected space countably cozero complemented space $z^0_c$-ideals |
| title | On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋) |
| title_full | On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋) |
| title_fullStr | On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋) |
| title_full_unstemmed | On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋) |
| title_short | On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋) |
| title_sort | on some properties of the space of minimal prime ideals of 𝐶𝑐 𝑋 |
| topic | the space of minimal prime ideals strongly zero-dimensional space countably basically disconnected space countably cozero complemented space $z^0_c$-ideals |
| url | https://cgasa.sbu.ac.ir/article_102622_fabfade2e239fe905af15ccfebc0a21e.pdf |
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