Algebraic characterizations of some disconnected spaces

It is shown that X is an extremally disconnected space if and only if C(X) is a Baer ring. We also give several new algebraic characterizations of basically disconnected spaces. These characterizations are then used to give a unified proof of the fact that X is extremally (basically) disconnected space if and only if 3X is extremally (basically) disconnected space. Zero-dimensional and strongly zero-dimensional spaces are also characterized similarly. It is shown that X is strongly zero-dimensional F-space if and only if each minimal prime ideal in C(X) is generated by idempotents. We also show that X is an extremally disconnected P-space with a dense set of isolated points if and only if C(X) is isomorphic to a direct product of fields. Finally, we prove that C(X) is a self injective ring if and only if X is an extremally disconnected P-space.