TITLE: Is Stone-Čech compactification of 0-dimensional space also 0-dimensional?
QUESTION [14 upvotes]: What is an example of a 0-dimensional locally compact Hausdorff space $X$ for which the Stone-Čech 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 $\mathrm{Cl}(X)$ of closed-open subsets is clearly a Boolean algebra extension of L, and the Stone space of $\mathrm{Cl}(X)$ 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 $L$.

REPLY [3 votes]: Thanks to KP.  This is the class $\Psi$ of Gillman-Jerison 5I, but apparently Terasawa explained more some 16 years later. 
There is a little history here. As pointed out by Figa-Talamanca and Franklin, "Multipliers of Distributive Lattices", Indian J. Math. 12 (1970) p. 159, John Kelley asserted in his topology book (p. 169) that the Cech-Stone compactification is always zero-dimensional.  They attempt to correct this error by citing an unspecified example in the 1955 paper of Dowker, "Local Dimension of Normal Spaces", Quart. J. Math. 2 (1955) 101-120, and they say this example was pointed out to them by J. Isbell and P. Dwinger.  However, I have not yet been able to verify that any of the non-strongly zero-dimensional examples in Dowker's paper are locally compact.  I wonder which one they had in mind?  If they were right, this would be a much earlier example than Terasawa's $\Psi$.  Could all four of them been mis-reading Dowker?
Here is a link to my copy of the Dowker paper. Does it contain a counterexample?