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An ideal I of a ring A is a z-ideal if whenever a, bA belong to the same maximal ideals of A and aI, then bI as well. On the other hand, an ideal J of A is a d-ideal if Ann2(a) ⊆ J for every aJ. 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 : LM 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.


This article was originally published in Mathematica Slovaca, volume 68, issue 2, in 2018.

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De Gruyter

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