More on the Functor Induced by z-Ideals

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An ideal I of a commutative ring A with identity is called a z-ideal if whenever two elements of A belong to the same maximal ideals and one of the elements is in I , then so is the other. For a completely regular frame L we denote by ZId(RL) the lattice of z-ideals of the ring RL of continuous real-valued functions on L. It is a coherent frame, and it is known that L → ZId(RL) is the object part of a functor Z : CRFrm → CohFrm, where CRFrm is the category of completely regular frames and frame homomorphisms, and CohFrm is the category of coherent frames and coherent maps. We explore when this functor preserves and reflects the property of being a Heyting homomorphism, and also when it preserves and reflects the variants of openness of Banaschewski and Pultr (Appl Categ Struct 2:331–350, 1994). We also record some other properties of this functor that have hitherto not been stated anywhere.


This article was originally published in Applied Categorical Structures, volume 26, in 2018.

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