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 |
| Summary: | 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 $L={bf2}$. |
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| ISSN: | 2345-5128 2345-511X |