What is an example of a 0-dimensional locally compact Hausdorff space X for which the Cech-Stone compactification beta(X) is NOT 0-dimensional?
It is known that if X is a 0-dimensional locally compact Hausdorff space which is also paracompact, then beta(X) is 0-dimensional. (Engelking 1989, Th. 6.2.9). I would expect a counterexample in the non-paracompact case.
Another way of asking the question is to look at a Boolean ring L (without assuming a unit). If X is the Stone space of L then X is a 0-dimensional locally compact Hausdorff space and L is isomorphic with the ring of compact open sets in X. The Boolean algebra Cl(X) of closed-open sets is clearly a Boolean algebra extension of L, and the Stone space of Cl(X) is a certain compactification of X. I believe it is easy to see that this compactification of X is just beta(X) if and only if beta(X) is 0-dimensional.
Thus the question can be expressed in the algebraic side of the duality. Find conditions on the Boolean ring L so that beta(X) is 0-dimensional, where X is the Stone space of L.
I think this may be true if, for example, L is a Boolean sigma-ring. My original question asks for an example showing this is not true without some additional conditions on L.