Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

We give characterizations of P-frames, essential P-frames and strongly zero-dimensional frames in terms of ring-theoretic properties of the ring of continuous real-valued functions on a frame. We define the m-topology on the ring RL and show that if L belongs to a certain class of frames properly containing the spatial ones, then L is a P-frame iff every ideal of RL is m-closed. We define essential P-frames (analogously to their spatial antecedents) and show that L is a proper essential P-frame iff all the nonmaximal prime ideals of RL are contained in one maximal ideal. Further, we show that L is strongly zero-dimensional iff RL is a clean ring, iff certain types of ideals of RL are generated by idempotents.