Purity of the ideal of continuous functions with compact support (Q2711342)

Let \(X\) be a completely regular Hausdorff space, \(C_K(X)\) denote the ring of all continuous real-valued functions on \(X\) with compact support, and \(X_L\) the set of points in \(X\) with compact neighbourhoods. We say that \(C_K(X)\) is pure in case, for each \(f\in C_K(X)\), there is \(g\in C_K(X)\) such that \(fg=f\). The author shows that \(C_K(X)\) is pure if, and only if, \(X_L\) is the union of supports of all \(f\in C_K(X)\). He extends some results of \textit{E. M. Vechtomov} [Russ. Math. Surv. 37, No. 4, 147-148 (1982); translation from Usp. Mat. Nauk 37, No. 4(226), 151-152 (1982; Zbl 0536.54009)] and others on the ideal structure of \(C_K(X)\) by assuming, not that \(X\) is locally compact as they did, but that \(C_K(X)\) is pure. One such result is that if \(C_K(X)\) and \(C_K(Y)\) are pure, then \(X_L\) and \(Y_L\) are homeomorphic if, and only if, \(C_K(X)\) and \(C_K(Y)\) are ring isomorphic. Another is that if \(X\) is locally compact, then \(X\) is basically disconnected if, and only if, every principal ideal of \(C_K(X)\) is a projective \(C(X)\)-module.