The epimorphic hull of \(C(X)\) (Q1570918)

Let \(X\) be a completely regular Hausdorff space and \(C(X)\) denote the ring of continuous real-valued functions with domain \(X\). The ring of quotients \(Q(X)\) is the ring of all continuous real-valued functions on the dense open subsets of \(X\), where functions are identified that agree on the intersection of their domains. In this very comprehensive survey, the authors study the epimorphic hull \(H(X)\) of \(C(X)\) defined as the smallest regular ring lying between \(C(X)\) and \(Q(X)\). Among other things, they show that (i) if \(Y\) is dense and \(C^*\)-embedded in \(X\), then \(H(Y)=H(X)\); (ii) if \(X\) is a \(P\)-space, then \(H(X)=C(X)\); (iii) if \(X\) is extremally disconnected, then \(H(X)= Q(X)\). They obtain various conditions on \(X\) in order that (iv) \(C(X)\) have a regular ring of quotients of the form \(C(Y)\); (v) \(H(X)\) have the form \(C(Y)\); (vi) the classical ring of quotients \(Q_{\text{cl}}(X)\) have the form \(C(Y)\). Examples are given to show that various coincidences between \(C(X)\subset Q_{\text{cl}}(X)\subset H(X)\subset Q(X)\) may or may not occur, and some open questions are raised.