Notes Concerning Characterizations of Quasi-F Frames

Abstract

We give several characterizations of quasi F-frames. To name a few, let ℛL denote, as usual, the ring of real-valued continuous functions on a completely regular frame L. Then, L is quasi-F iff Ann²(α) + Ann²(β) = ℛL whenever α + β is not a zero-divisor. A commutative ring is said to be quasi-Bézout if every finitely generated ideal that contains a non-zero-divisor is principal. We show that L is quasi-F iff ℛL is quasi-Bézout. Further, L is quasi-F iff βL (its Stone-Čech compactification) is quasi-F, iff λL (its universal Lindelőfication) is quasi-F, iff vL (its Hewitt realcompactification) is quasi-F. Element-wise characterizations include one stating that a completely regular frame is quasi-F iff whenever the join of two cozero elements is dense, then their pseudocomplements are completely separated.

Key words:

• Frame
• quasi F-frame
• ring of real-valued continuous functions on a frame