An extension of $F$-spaces and its applications

A completely regular Hausdorff space $X$ is called a $WCF$-space if every pair of disjoint cozero-sets in $X$ can be separated by two disjoint $Z^{\circ}$-sets. The class of $WCF$-spaces properly contains both the class of $F$-spaces and the class of cozero-complemented spaces. We prove that if $Y$ is a dense $z$-embedded subset of a space $X$, then $Y$ is a $WCF$-space if and only if $X$ is a $WCF$-space. As a consequence, a completely regular Hausdorff space $X$ is a $WCF$-space if and only if $\beta X$ is a $WCF$-space if and only if $\upsilon X$ is a $WCF$-space. We then apply this concept to introduce the notions of $PW$-rings and $UPW$-rings. A ring $R$ is called a $PW$-ring (resp., $UPW$-ring) if for all $a, b \in R$ with $aR \cap bR = 0$, the ideal $\Ann(a)+\Ann(b)$ contains a regular element (resp., a unit element). It is shown that $C(X)$ is a $PW$-ring if and only if $X$ is a $WCF$-space, if and only if $C^{*}(X)$ is a $PW$-ring. Moreover, for a reduced $f$-ring $R$ with bounded inversion, we prove that the lattice $BZ^{\circ}(R)$ is co-normal if and only if $R$ is a $PW$-ring. Several examples are provided to illustrate and delimit our results.