Notes on Pointfree Disconnectivity with a Ring-theoretic Slant

Abstract

We give characterizations of extremally disconnected frames, basically disconnected frames and F-frames L in terms of ring-theoretic properties of the ring \(\mathcal{R}L\) of continuous real-valued functions on L. Emanating from these are new (and purely ring-theoretic) proofs that a frame is extremally disconnected, basically disconnected or an F-frame iff the same holds for its Cech-Stone compactification.