Cozero complemented frames

When certain prime ideals in rings of continuous functions are minimal or maximal

“…In [2, Proposition 1.26] the authors show, among other things, that if a ring R has Property A, then every prime d-ideal of R is minimal prime if and only if for every a ∈ R there exists b ∈ R such that Ann(a) = Ann 2 (b). Proposition 1.1 in [19] shows that L is cozero complemented if and only if for every α ∈ RL there is a β ∈ RL such that Ann(α) = Ann 2 (β). Now let us show that RL has Property A.…”

“…We do however have a class of frames for which this can be asserted. Following [19], we say a point I of βL is sharp if, for any c ∈ Coz L, r L (c) ≤ I implies c ∈ I. We say it is almost sharp if, for any c ∈ Coz L, r L (c) ≤ I implies c is not dense.…”

“…We say it is almost sharp if, for any c ∈ Coz L, r L (c) ≤ I implies c is not dense. It is shown in [19] that L is a P -frame (resp. almost P -frame) precisely when every point of βL is sharp (resp.…”