A Tychonoff space X is called a quasi m-space if every prime d-ideal of the ring C(X) is either a maximal ideal or a minimal prime ideal. These spaces were characterised by Azarpanah and Karavan. In this paper we look at some properties of these spaces from a ring-theoretic perspective. We observe, for instance, that among subspaces which inherit this property are (i) cozero subspaces, (ii) dense z-embedded subspaces, and (iii) regular-closed subspaces among the normal quasi m-spaces. The ring-theoretic approach actually yields the above results within the broader context of frames. The latter part of the paper discusses completely regular frames L for which every prime z-ideal in the ring RL is a maximal ideal or a minimal prime ideal.
Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.
We define weakly regular rings by a condition characterizing the rings C(X) for weak almost P-spaces X. A Tychonoff space X is called a weak almost P-space if for every two zero-sets E and F of X with int E ⊆ int F, there is a nowhere dense zero-set H of X such that E ⊆ F ∪ H. We show that a reduced f -ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. Frames L for which the ring RL of real-valued continuous functions on L is weakly regular are characterized. We show that if the coproduct of two Lindel öf frames is of this kind, then so is each summand. Also, a continuous Lindel öf frame is of this kind if and only if its Stone-Čech compactification is of this kind.
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