Concerning P-frames and the Artin-Rees property

Let RL be the ring of continuous real-valued functions on a completely regular frame L. The Artin-Rees property in RL, in the factor rings of RL and in the rings of fractions of RL is studied. We show that a frame L is a P-frame if and only if RL is an Artin-Rees ring if and only if every ideal of RL with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring RL/<phi > is an Artin-Rees ring for any phi is an element of RL. 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 are an Artin-Rees ring if and only if L is a P-frame. We show that the complete ring of fractions of RL 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 is compact. Finally, we prove that if phi is an element of RL such that the open quotient down arrow coz phi is a dense C-quotient of L, then the ring of fractions (RL)(phi) is regular if and only if down arrow coz phi is a P-frame.