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Keywords:
frame; ring of real-valued continuous function; strongly zero-dimensional; clean element; sublocale
frame; ring of real-valued continuous function; strongly zero-dimensional; clean element; sublocale
Summary:
We characterize clean elements of $\mathcal R(L)$ and show that $\alpha \in \mathcal {R}(L)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that $\frak {c}_L({\rm coz} (\alpha - {\bf 1})) \subseteq U \subseteq \frak {o}_L( {\rm coz} (\alpha ))$. Also, we prove that $\mathcal R(L)$ is clean if and only if $\mathcal R(L)$ has a clean prime ideal. Then, according to the results about $\mathcal R(L),$ we immediately get results about $\mathcal C_{c}(L).$
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