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When spectra of lattices of \(z\)-ideals are Stone-Čech compactifications. (English) Zbl 1463.06042

Summary: Let \(X\) be a completely regular Hausdorff space and, as usual, let \(C(X)\) denote the ring of real-valued continuous functions on \(X\). The lattice of \(z\)-ideals of \(C(X)\) has been shown by J. Martínez and E. R. Zenk [Commentat. Math. Univ. Carol. 46, No. 4, 607–636 (2005; Zbl 1121.06009)] to be a frame. We show that the spectrum of this lattice is (homeomorphic to) \(\beta X\) precisely when \(X\) is a \(P\)-space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a \(d\)-ideal if whenever two elements have the same annihilator and one of the elements belongs to the ideal, then so does the other. We characterize when the spectrum of the lattice of \(d\)-ideals of \(C(X)\) is the Stone-Čech compactification of the largest dense sublocale of the locale determined by \(X\). It is precisely when the closure of every open set of \(X\) is the closure of some cozero-set of \(X\).

MSC:

06D22 Frames, locales
54E17 Nearness spaces
13A15 Ideals and multiplicative ideal theory in commutative rings
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)

Citations:

Zbl 1121.06009
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