On $z$-ideals of pointfree function rings

Document Type: Research Paper

Authors

1 University of South Africa

2 University of South Africa (Unisa)

Abstract

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ‎ring of continuous real-valued functions on $L$‎. ‎We show that the‎ ‎lattice $Zid(mathcal{R}L)$ of $z$-ideals of $mathcal{R}L$ is a‎ ‎normal coherent Yosida frame‎, ‎which extends the corresponding $C(X)$‎ ‎result of Mart'{i}nez and Zenk‎. ‎This we do by exhibiting‎ ‎$Zid(mathcal{R}L)$ as a quotient of $Rad(mathcal{R}L)$‎, ‎the‎ ‎frame of radical ideals of $mathcal{R}L$‎. ‎The saturation quotient‎ ‎of $Zid(mathcal{R}L)$ is shown to be isomorphic to the‎ ‎Stone-v{C}ech compactification of $L$‎. ‎Given a morphism $hcolon ‎Lto M$ in $mathbf{CRegFrm}$‎, ‎$Zid$ creates a coherent frame‎ ‎homomorphism $Zid(h)colonZid(mathcal{R}L)toZid(mathcal{R}M)$‎ ‎whose right adjoint maps as $(mathcal{R}h)^{-1}$‎, ‎for the induced‎ ‎ring homomorphism $mathcal{R}hcolonmathcal{R}Ltomathcal{R}M$‎.‎Thus‎, ‎$Zid(h)$ is an $s$-map‎, ‎in the sense of Mart`{i}nez cite{Mar1}‎, ‎precisely when‎ ‎$mathcal{R}(h)$ contracts maximal ideals to maximal ideals‎.

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