Let RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} be the ring of continuous real-valued functions on a completely regular frame L. The Artin–Rees property in RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document}, in the factor rings of RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} and in the rings of fractions of RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} is studied. We show that a frame L is a P-frame if and only if RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} is an Artin–Rees ring if and only if every ideal of RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} with the Artin–Rees property is an Artin–Rees ideal if and only if the factor ring RL/⟨φ⟩\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L/\langle \varphi \rangle $$\end{document} is an Artin–Rees ring for any φ∈RL\documentclass[12pt]{minimal}
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\begin{document}$$\varphi \in {\mathcal {R}}L$$\end{document}. A necessary and sufficient condition for the local rings of a reduced ring to be Artin–Rees rings is that each of its prime ideals becomes minimal. It turns out that the local rings of RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} are an Artin–Rees ring if and only if L is a P-frame. We show that the complete ring of fractions of RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} is an Artin–Rees ring if and only if L is a cozero-complemented frame, or equivalently, the set of all minimal prime ideals of the ring RL\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}L$$\end{document} is compact. Finally, we prove that if φ∈RL\documentclass[12pt]{minimal}
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\begin{document}$$\varphi \in {\mathcal {R}}L$$\end{document} such that the open quotient ↓cozφ\documentclass[12pt]{minimal}
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\begin{document}$$\downarrow \!\!{{\,\mathrm{coz}\,}}\varphi $$\end{document} is a dense C-quotient of L, then the ring of fractions (RL)φ\documentclass[12pt]{minimal}
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\begin{document}$$({\mathcal {R}}L)_\varphi $$\end{document} is regular if and only if ↓cozφ\documentclass[12pt]{minimal}
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\begin{document}$$\downarrow \!\!{{\,\mathrm{coz}\,}}\varphi $$\end{document} is a P-frame.
