A minimal regular ring extension of $C(X)$
Let $G(X)$ denote the smallest (von Neumann) regular ring of real-valued functions with domain $X$ that contains $C(X)$, the ring of continuous real-valued functions on a Tikhonov topological space $(X, \tau )$. We investigate when $G(X)$ coincides with the ring $C(X, \tau _\delta )$ of continuous real-valued functions on the space $(X, \tau _\delta )$, where $\tau _\delta $ is the smallest Tikhonov topology on $X$ for which $\tau \subseteq \tau _\delta $ and $C(X, \tau _\delta )$ is von Neumann regular. The compact and metric spaces for which $G(X) = C(X, \tau _\delta )$ are characterized. Necessary, and different sufficient, conditions for the equality to hold more generally are found.