More on the Functor Induced by z-Ideals

摘要

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 the lattice of z-ideals of the ring of continuous real-valued functions on L. It is a coherent frame, and it is known that is the object part of a functor , where is the category of completely regular frames and frame homomorphisms, and 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.