When rings of continuous functions are weakly regular

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.