## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### When rings of continuous functions are weakly regular

Themba Dube and Jissy Nsonde Nsayi

#### 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