ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS
We study primary ideals of the ring $mathcal{R}L$ of real-valued continuous functions on a completely regular frame $L$. We observe that prime ideals and primary ideals coincide in a $P$-frame. It is shown that every primary ideal in $mathcal{R}L$ is contained in a unique maximal ideal, and an ideal...
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| Format: | Article |
| Language: | English |
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Shahrood University of Technology
2020-01-01
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| Series: | Journal of Algebraic Systems |
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| Online Access: | http://jas.shahroodut.ac.ir/article_1594_8108dcd6f42553890dfb9b0ba2055003.pdf |
| _version_ | 1819211581250076672 |
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| author | M. Abedi |
| author_facet | M. Abedi |
| author_sort | M. Abedi |
| collection | DOAJ |
| description | We study primary ideals of the ring $mathcal{R}L$ of real-valued continuous functions on a completely regular frame $L$. We observe that prime ideals and primary ideals coincide in a $P$-frame. It is shown that every primary ideal in $mathcal{R}L$ is contained in a unique maximal ideal, and an ideal $Q$ in $mathcal{R}L$ is primary if and only if $Q capmathcal{R}^*L$ is a primary ideal in $mathcal{R}^*L$. We show that every pseudo-prime (primary) ideal in $mathcal{R}L$ is either an essential ideal or a maximal ideal which is at the same time a minimal prime ideal. Finally, we prove that if $L$ is a connected frame, then the zero ideal in $mathcal{R}L$ is decomposable if and only if $L={bf2}$. |
| first_indexed | 2024-12-23T06:29:21Z |
| format | Article |
| id | doaj.art-8080dcb50f5f4283b5e46a63f88f4d02 |
| institution | Directory Open Access Journal |
| issn | 2345-5128 2345-511X |
| language | English |
| last_indexed | 2024-12-23T06:29:21Z |
| publishDate | 2020-01-01 |
| publisher | Shahrood University of Technology |
| record_format | Article |
| series | Journal of Algebraic Systems |
| spelling | doaj.art-8080dcb50f5f4283b5e46a63f88f4d022022-12-21T17:56:58ZengShahrood University of TechnologyJournal of Algebraic Systems2345-51282345-511X2020-01-017225726910.22044/jas.2019.8150.13991594ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGSM. Abedi0Esfarayen University of Technology, Esfarayen, North Khorasan, Iran.We study primary ideals of the ring $mathcal{R}L$ of real-valued continuous functions on a completely regular frame $L$. We observe that prime ideals and primary ideals coincide in a $P$-frame. It is shown that every primary ideal in $mathcal{R}L$ is contained in a unique maximal ideal, and an ideal $Q$ in $mathcal{R}L$ is primary if and only if $Q capmathcal{R}^*L$ is a primary ideal in $mathcal{R}^*L$. We show that every pseudo-prime (primary) ideal in $mathcal{R}L$ is either an essential ideal or a maximal ideal which is at the same time a minimal prime ideal. Finally, we prove that if $L$ is a connected frame, then the zero ideal in $mathcal{R}L$ is decomposable if and only if $L={bf2}$.http://jas.shahroodut.ac.ir/article_1594_8108dcd6f42553890dfb9b0ba2055003.pdfframeprimary idealpseudo-prime idealring of continuous real-valued functionsdecomposable ideal |
| spellingShingle | M. Abedi ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS Journal of Algebraic Systems frame primary ideal pseudo-prime ideal ring of continuous real-valued functions decomposable ideal |
| title | ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS |
| title_full | ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS |
| title_fullStr | ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS |
| title_full_unstemmed | ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS |
| title_short | ON PRIMARY IDEALS OF POINTFREE FUNCTION RINGS |
| title_sort | on primary ideals of pointfree function rings |
| topic | frame primary ideal pseudo-prime ideal ring of continuous real-valued functions decomposable ideal |
| url | http://jas.shahroodut.ac.ir/article_1594_8108dcd6f42553890dfb9b0ba2055003.pdf |
| work_keys_str_mv | AT mabedi onprimaryidealsofpointfreefunctionrings |