Some algebraic characterizations of F-frames

Abstract

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.

Covering maximal ideals with minimal primes

“…Next, we recall from [16] that a collection F of minimal prime ideals of RL is said to be adequate for a quotient map h : βL → M if for every I ∈ Pt(βL) with h(I) < 1, there exists Q ∈ F such that Q ⊆ M I . We wish to cast this definition in a slightly different, but equivalent, way that will make it easier for our purposes to work with.…”

“…In [16], it is shown that every round quotient map βL → 2 is almost round. The following proposition strengthens this.…”

“…A P -frame is a frame in which every cozero element is complemented. We refer to [16] and [15] for some characterizations of F -frames and P -frames, respectively.…”

“…Since L is an F -frame, [16,Proposition 3.11] tells us that ξ is almost round. But by the current hypothesis, it is also nearly round, so it is round by the foregoing proposition because its codomain is spatial.…”

“…We extend this concept in Section 3. In [13] and [16], respectively, round and almost round quotient maps were introduced. In the latter paper, it was shown that every round quotient map into 2, the two-element frame, is almost round.…”