More ring-theoretic characterizations of P-frames

Call a ring z-good if it has the property that an ideal in it is a z-ideal if and only if its radical is a z-ideal. By first showing that a z-good ring is von Neumann regular if and only if every prime ideal in it is a z-ideal, we are able to characterize P-frames as precisely those L for which every prime ideal in the ring [Formula: see text] is a z-ideal. Furthermore, we show that this characterization still holds if prime ideals are replaced by essential ideals, radical ideals, convex ideals, or absolutely convex ideals.