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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.05473 |
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| _version_ | 1866916352731643904 |
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| author | Abedi, Mostafa |
| author_facet | Abedi, Mostafa |
| contents | Consider the subring $\mathcal{R}_cL$ of continuous real-valued functions defined on a frame $L$, comprising functions with a countable pointfree image. We present some useful properties of $\mathcal{R}_cL$. We establish that both $\mathcal{R}_cL$ and its bounded part, $\mathcal{R}_c^*L$, are clean rings for any frame $L$. We show that, for any completely regular frame $L$, the $z_c$-ideals of $\mathcal{R}_cL$ are contractions of the $z$-ideals of $\mathcal{R}L$. This leads to the conclusion that maximal ideals (or prime $z_c$-ideals) of $\mathcal{R}_cL$ correspond precisely to the contractions of those of $\mathcal{R}L$. We introduce the ${\bf O}_c$- and ${\bf M}_c$-ideals of $\mathcal{R}_cL$. By using ${\bf M}_c$-ideals, we characterize the maximal ideals of $\mathcal{R}_cL$, drawing an analogy with the Gelfand-Kolmogoroff theorem for the maximal ideals of $C_c(X)$. We demonstrate that fixed maximal ideals of $\mathcal{R}_cL$ have a one-to-one correspondence with the points of $L$ in the case where $L$ is a zero-dimensional frame. We describe the maximal ideals of $\mathcal{R}_c^*L$, leading to a one-to-one correspondence between these ideals and the points of $βL$, the Stone-Čech compactification of $L$, when $L$ is a strongly zero-dimensional frame. Finally, we establish that $β_0L$, the Banaschewski compactification of a zero-dimensional $L$, is isomorphic to the frames of the structure spaces of $\mathcal{R}_cL$, $\mathcal{R}_c(β_0L)$, and $\mathcal{R}(β_0L)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal Ideals in Functions Rings with a Countable Pointfree Image Abedi, Mostafa Functional Analysis Rings and Algebras Primary 06D22, Secondary 54C30, 13A15, 54C40, 06B10 Consider the subring $\mathcal{R}_cL$ of continuous real-valued functions defined on a frame $L$, comprising functions with a countable pointfree image. We present some useful properties of $\mathcal{R}_cL$. We establish that both $\mathcal{R}_cL$ and its bounded part, $\mathcal{R}_c^*L$, are clean rings for any frame $L$. We show that, for any completely regular frame $L$, the $z_c$-ideals of $\mathcal{R}_cL$ are contractions of the $z$-ideals of $\mathcal{R}L$. This leads to the conclusion that maximal ideals (or prime $z_c$-ideals) of $\mathcal{R}_cL$ correspond precisely to the contractions of those of $\mathcal{R}L$. We introduce the ${\bf O}_c$- and ${\bf M}_c$-ideals of $\mathcal{R}_cL$. By using ${\bf M}_c$-ideals, we characterize the maximal ideals of $\mathcal{R}_cL$, drawing an analogy with the Gelfand-Kolmogoroff theorem for the maximal ideals of $C_c(X)$. We demonstrate that fixed maximal ideals of $\mathcal{R}_cL$ have a one-to-one correspondence with the points of $L$ in the case where $L$ is a zero-dimensional frame. We describe the maximal ideals of $\mathcal{R}_c^*L$, leading to a one-to-one correspondence between these ideals and the points of $βL$, the Stone-Čech compactification of $L$, when $L$ is a strongly zero-dimensional frame. Finally, we establish that $β_0L$, the Banaschewski compactification of a zero-dimensional $L$, is isomorphic to the frames of the structure spaces of $\mathcal{R}_cL$, $\mathcal{R}_c(β_0L)$, and $\mathcal{R}(β_0L)$. |
| title | Maximal Ideals in Functions Rings with a Countable Pointfree Image |
| topic | Functional Analysis Rings and Algebras Primary 06D22, Secondary 54C30, 13A15, 54C40, 06B10 |
| url | https://arxiv.org/abs/2408.05473 |