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Paper Information

Journal:   BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY   2014 , Volume 40 , Number 3; Page(s) 657 To 675.
 
Paper: 

ON Z-IDEALS OF POINTFREE FUNCTION RINGS

 
 
Author(s):  DUBE THEMBA*, IGHEDO OGHENETEGA
 
* DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF SOUTH AFRICA, PRETORIA, SOUTH AFRICA
 
Abstract: 

Let L be a completely regular frame and RL be the ring of continuous real-valued functions on L. We show that the lattice Zid (RL) of z-ideals of RL is a normal coherent Yosida frame, which extends the corresponding C (X) result of Martinez and Zenk. This we do by exhibiting Zid (RL) as a quotient of Rad (RL), the frame of radical ideals of RL. The saturation quotient of Zid (RL) is shown to be isomorphic to the Stone-Cech compactification of L. Given a morphism h: L®M in CRegFrm, Zid creates a coherent frame homomorphism Zid (h): Zid (RL)®Zid (RM) whose right adjoint maps as (Rh)-1, for the induced ring homomorphism Rh: RL®RM. Thus, Zid (h) is an s-map, in the sense of Martinez [18], precisely when R (h) contracts maximal ideals to maximal ideals.

 
Keyword(s): FRAME, IDEAL, Z-IDEAL
 
References: 
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