Some algebraic characterizations of F-frames

In pointfree topology, F-frames have been defined by Ball and Walters-Wayland 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 ${mathcal{R}L}$ 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 ${mathcal{R}L}$ is almost weak Baer iff L is a strongly zero-dimensional F-frame.