Almost P -frames generalize almost P -spaces, and indeed transcend them. In this article we give several characterizations of these frames, mostly in terms of certain ring-theoretic properties of RL, the ring of real-valued functions on L.

A frame homomorphism is coz-onto if it maps the cozero part of its domain surjectively onto that of its codomain. This captures the notion of a z-embedded subspace of a topological space in a point-free setting. We give three different types of characterizations of coz-onto homomorphisms. The first is in terms of elements, the second in terms of quotients, and the last in terms of ideals. As an application of properties of coz-onto homomorphisms developed herein, we present some characterizations of F-and F -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 Pframe 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.

We study a property of frames which is akin to realcompactness and obtained by replacing the cozero part of a frame in the definition of realcompactness with its Booleanization. Unlike the case of realcompactness, which is defined only for completely regular frames, this new concept is defined for all frames. We also investigate a weaker variant of this notion, and note that in both cases the frame results extend their topological precursors.

In pointfree topology, F -frames have been defined by Ball and WaltersWayland by means of a frame-theoretic translation of the topological characterization of F -spaces as those whose cozero-sets are C * -embedded. This is a departure from the way in which F -spaces were defined by Gillman and Henriksen as those spaces X for which the ring C(X) is Bézout, meaning that every finitely generated ideal is principal. In this note, we show that, as in the case of spaces, a frame L is an F -frame precisely when the ring RL of continuous real-valued functions on L is Bézout. A commutative ring with identity is called almost weak Baer if the annihilator of each element is generated by idempotents. We establish that RL is almost weak Baer iff L is a strongly zero-dimensional F -frame.