Let A be a reduced commutative f-ring with identity and bounded inversion, and let A* be its subring of bounded elements. By first observing that A is the ring of fractions of A* relative to the subset of A* consisting of elements which are units in the bigger ring, we show that the frames Did (A) and Did (A*) of d-ideals of A and A*, respectively, are isomorphic, and that the isomorphism witnessing this is precisely the restriction of the extension map I ↦ Ie which takes a radical ideal of A* to the ideal it generates in A. Specializing to the ring [Formula: see text], we show that if L is an F-frame, then the saturation quotient of [Formula: see text] is isomorphic to βL. We also investigate projectability properties of [Formula: see text] and [Formula: see text], where the latter denotes the frame of z-ideals of [Formula: see text]. We show that [Formula: see text] is flatly projectable precisely when [Formula: see text] is a feebly Baer ring. Quite easily, [Formula: see text] is projectable if and only if L is basically disconnected. Less obvious is that [Formula: see text] is projectable if and only if L is cozero-complemented.

An ideal I of a ring A is a z-ideal if whenever a, b ∈ A belong to the same maximal ideals of A and a ∈ I, then b ∈ I as well. On the other hand, an ideal J of A is a d-ideal if Ann2(a) ⊆ J for every a ∈ J. It is known that the lattices Z(L) and D(L) of the ring 𝓡L of continuous real-valued functions on a frame L, consisting of z-ideals and d-ideals of 𝓡L, respectively, are coherent frames. In this paper we characterize, in terms of the frame-theoretic properties of L (and, in some cases, the algebraic properties of the ring 𝓡L), those L for which Z(L) and D(L) satisfy the various regularity conditions on algebraic frames introduced by Martínez and Zenk [20]. Every frame homomorphism h : L → M induces a coherent map Z(h) : Z(L) → Z(M). Conditions are given of when this map is closed, or weakly closed in the sense Martínez [19]. The case of openness of this map was discussed in [11]. We also prove that, as in the case of the ring C(X), the sum of two z-ideals of 𝓡L is a z-ideal.

No abstract

No abstract

No abstract