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.