Resumen de Concerning P-frames and the Artin–Rees property
Let {\mathcal {R}}L be the ring of continuous real-valued functions on a completely regular frame L. The Artin–Rees property in {\mathcal {R}}L, in the factor rings of {\mathcal {R}}L and in the rings of fractions of {\mathcal {R}}L is studied. We show that a frame L is a P-frame if and only if {\mathcal {R}}L is an Artin–Rees ring if and only if every ideal of {\mathcal {R}}L with the Artin–Rees property is an Artin–Rees ideal if and only if the factor ring {\mathcal {R}}L/\langle \varphi \rangle is an Artin–Rees ring for any \varphi \in {\mathcal {R}}L. 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 {\mathcal {R}}L are an Artin–Rees ring if and only if L is a P-frame. We show that the complete ring of fractions of {\mathcal {R}}L 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 {\mathcal {R}}L is compact. Finally, we prove that if \varphi \in {\mathcal {R}}L such that the open quotient \downarrow \!\!{{\,\mathrm{coz}\,}}\varphi is a dense C-quotient of L, then the ring of fractions ({\mathcal {R}}L)_\varphi is regular if and only if \downarrow \!\!{{\,\mathrm{coz}\,}}\varphi is a P-frame.
