Bulletin of the Belgian Mathematical Society - Simon Stevin

When rings of continuous functions are weakly regular

Themba Dube and Jissy Nsonde Nsayi

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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

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

First available in Project Euclid: 28 May 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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