Abstract
We prove that the lattice of z-ideals of a commutative ring with identity is a coherent frame. We characterize when it is a Yosida frame, and when it satisfies some projectability properties. We also characterize Hilbert rings in terms of ideals that arise naturally in this study. A ring with zero Jacobson radical is shown to be feebly Baer precisely when its frame of z-ideals is feebly projectable. Denote by ZId(A) the frame of z-ideals of a ring A. We show that the assignment A↦ZId(A) is the object part of a functor CRngz→CohFrm, where CRngz designates the category whose objects are commutative rings with identity and whose morphisms are the ring homomorphisms that contract z-ideals to z-ideals.
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