ON CP-FRAMES AND THE ARTIN-REES PROPERTY

The set Cc(L) = { α ∈ RL : |{r ∈ R : coz(α − r) ≠ 1}| ≤ ℵ0 o is a sub-f-ring of RL, that is, the ring of all continuous real-valued functions on a completely regular frame L. The main purpose of this paper is to continue our investigation begun in [3] of extending ring-theoretic properties in RL to the context of completely regular frames by replacing the ring RL with the ring Cc(L) to the context of zero-dimensional frames. We show that a frame L is a CP-frame if and only if Cc(L) is a regular ring if and only if every ideal of Cc(L) is pure if and only if Cc(L) is an Artin-Rees ring if and only if every ideal of Cc(L) with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring Cc(L)/hαi is an Artin-Rees ring for any α ∈ Cc(L) if and only if every minimal prime ideal of Cc(L) is an Artin-Rees ideal.