Contracting the socle in rings of continuous functions
The socle of a ring is the ideal generated by the minimal ideals of the ring. This paper starts with the classical result that for a completely regular Hausdorff space \(X\), the socle of the ring \(C(X)\) is the ideal consisting of all functions that are zero everywhere except on a finite number of points. The goal of the paper is to study the socle of the ring \({\mathcal R}L\) of real continuous functions on a frame (locale) \(L\). The author starts by characterizing the socle of \({\mathcal R}L\): it is the ideal consisting of all functions whose cozero elements are finite joins of atoms of \(L\). The spatial result above follows as an easy corollary. It also follows that for any surjective frame homomorphism \(h:L\rightarrow M\), the induced ring homomorphism \({\mathcal R}h:{\mathcal R}L\rightarrow {\mathcal R}M\) extends the socle to an ideal contained in the socle (that is, \(({\mathcal R}h)[\mathrm{Soc}\, {\mathcal R}L]\subseteq \mathrm{Soc}\, {\mathcal R}M\)). Then it is proved that \(\mathrm{Soc}\, {\mathcal R}L\) is an exoteric ideal and that \({\mathcal R}h\) is exoteric for every nearly open frame homomorphism \(h\). Finally, a class of frame homomorphisms, called W-maps, is introduced. These maps characterize \(P\)-frames as those \(L\) for which every frame homomorphism with domain \(L\) is of this kind. Every dense W-map \(h:L\rightarrow M\) induces a ring homomorphism \({\mathcal R}h\) that contracts the socle of its codomain to an ideal contained in the socle of its domain (that is, \(({\mathcal R}h)^{-1}[\mathrm{Soc}\, {\mathcal R}M]\subseteq \mathrm{Soc} \, {\mathcal R}L\)). In case \(h\) is also surjective, \({\mathcal R}h\) contracts the socle of \({\mathcal R}M\) to the socle of \({\mathcal R}L\) (that is, \(({\mathcal R}h)^{-1}[\mathrm{Soc}\, {\mathcal R}M]= \mathrm{Soc} \, {\mathcal R}L\)).
scientific article; zbMATH DE number 5780243
scientific article