Wallman rings, disconnections and sub-direct sums

A subring A of C(X), the ring of continuous real functions on X, is said to be a Wallman ring if Z[A], the family of zero-sets of the elements of A, is a normal base. If a Wallman ring A consists of bounded functions, then one has two compactifications associated to A : the Wallman-type compactification w(Z(A)) and the compactification e_A(X) obtained by mean of the diagonal map e_A. In general, these two compactifications are not equivalent. In this paper we investigate the relationship between algebraic properties of a Wallman ring A and the disconnectedness of e_A(X) and w(Z(A)). In particular, we prove that w(Z(A)) is disconnected if and only if A is a sub-direct sum of two non-zero sub-rings S, T of C(X), such that their direct sum is a Wallman ring equivalent to A. A similar result for e_A(X) is proved. Also, some examples are given to show the indipendence between the disconnectedness of e_A(X) and w(Z(A)).