Purity of the ideal of continuous functions with pseudocompact support (Q1607865)

Summary: Let \(C_\Psi(X)\) be the ideal of functions with pseudocompact support and let \(kX\) be the set of all points in \(\upsilon X\) having compact neighborhoods. We show that \(C_\Psi(X)\) is pure if and only if \(\beta X-kX\) is a round subset of \(\beta X, C_\Psi(X)\) is a projective \(C(X)\)-module if and only if \(C_\Psi(X)\) is pure and \(kX\) is paracompact. We also show that if \(C_\Psi(X)\) is pure, then for each \(f \in C_\Psi(X)\) the ideal \((f)\) is a projective (flat) \(C(X)\)-module if and only if \(kX\) is basically disconnected (\(F'\)-space).