On ideals of rings of continuous functions associated with sublocales
| Autor: | Themba Dube, Dorca Nyamusi Stephen |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Ring (mathematics)
Mathematics::Commutative Algebra Nowhere dense set Tychonoff space 010102 general mathematics Join (topology) Lattice (discrete subgroup) 01 natural sciences 010101 applied mathematics Combinatorics Dense-in-itself Geometry and Topology Ideal (ring theory) 0101 mathematics Mathematics Complement (set theory) |
| Zdroj: | Topology and its Applications. 284:107360 |
| ISSN: | 0166-8641 |
| DOI: | 10.1016/j.topol.2020.107360 |
| Popis: | Let X be a Tychonoff space. Associated with every subset A ⊆ β X is the ideal O A of the ring C ( X ) consisting of all functions that vanish in a neighborhood of X ∩ A . Now, viewing Tychonoff spaces as objects in the category CRLoc of completely regular locales, we have ideals of the form O A , where A is a sublocale of βX. In this paper we study properties of such ideals not only for Tychonoff spaces, but for any object in CRLoc. Carrying out the discussion in this category, we have more function rings (they are denoted R L , for any L ∈ CRLoc ) than the class of the rings C ( X ) . Pure ideals of C ( X ) are known to be exactly the ideals O A , for A a closed subset of βX. We characterize the spaces for which the ideals O A , for A a closed subset of X (note that it need not be closed in βX) are pure ideals. They properly contain the normal ones. We describe the socle of any ring R L as the ideal O A , with A equal to the join of all nowhere dense sublocales of βL. We show that the socle is zero precisely when βL is dense in itself, and essential if and only if the smallest dense sublocale of βL has a complement in the lattice of sublocales of βL. If βL is scattered, then this is so if and only if L has a smallest nowhere dense sublocale. |
| Databáze: | OpenAIRE |
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