Bulletin of the Belgian Mathematical Society - Simon Stevin

When rings of continuous functions are weakly regular

Themba Dube and Jissy Nsonde Nsayi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We define weakly regular rings by a condition characterizing the rings $C(X)$ for weak almost $P$-spaces $X$. A Tychonoff space $X$ is called a weak almost $P$-space if for every two zero-sets $E$ and $F$ of $X$ with $\text{int } E\subseteq \text{int } F$, there is a nowhere dense zero-set $H$ of $X$ such that $E\subseteq F\cup H$. We show that a reduced $f$-ring is weakly regular if and only if every prime $z$-ideal in it which contains only zero-divisors is a $d$-ideal. Frames $L$ for which the ring $\mathcal{R}L$ of real-valued continuous functions on $L$ is weakly regular are characterized. We show that if the coproduct of two Lindelöf frames is of this kind, then so is each summand. Also, a continuous Lindelöf frame is of this kind if and only if its Stone-Čech compactification is of this kind.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 2 (2015), 213-226.

Dates
First available in Project Euclid: 28 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1432840859

Digital Object Identifier
doi:10.36045/bbms/1432840859

Mathematical Reviews number (MathSciNet)
MR3351037

Zentralblatt MATH identifier
1325.06009

Subjects
Primary: 06D22: Frames, locales {For topological questions see 54-XX}
Secondary: 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.) 54D60: Realcompactness and realcompactification

Keywords
frame weak almost $P$-frame Lindelöf frame $f$-ring weakly regular ring

Citation

Dube, Themba; Nsayi, Jissy Nsonde. When rings of continuous functions are weakly regular. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 2, 213--226. doi:10.36045/bbms/1432840859. https://projecteuclid.org/euclid.bbms/1432840859


Export citation