Abstract
We characterize clean elements of \({\cal R}(L)\) and show that \(\alpha \in {\cal R}(L)\) is clean if and only if there exists a clopen sublocale U in L such that \({\mathfrak{c}_L}({\rm{coz}}(\alpha - {\bf{1}})) \subseteq U \subseteq {_L}({\rm{coz}}(\alpha))\). Also, we prove that \({\cal R}(L)\) is clean if and only if \({\cal R}(L)\) has a clean prime ideal. Then, according to the results about \({\cal R}(L)\), we immediately get results about \({{\cal C}_c}(L)\).
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We would like to express our deep gratitude to the referee for improving the article with useful comments.
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Estaji, A.A., Taha, M. The clean elements of the ring \({\cal R}(L)\). Czech Math J 74, 211–230 (2024). https://doi.org/10.21136/CMJ.2023.0062-23
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DOI: https://doi.org/10.21136/CMJ.2023.0062-23