A correspondence between ideals and z-filters for certain rings of continuous functions – some remarks

Let X be a completely regular Hausdorff topological space and A ( X ) a ring lying between C ⁎ ( X ) and C ( X ) . A correspondence Z A between ideals of A ( X ) and the z-filters on X was initiated by Redlin and Watson in 1987 and was further investigated by Byun and Watson in a paper published in Topology and its Applications in 1991. In the last mentioned paper, the authors have established a lemma which reads that for any two rings A ( X ) and B ( X ) lying between C ⁎ ( X ) and C ( X ) with B ( X ) ⊆ A ( X ) and for any ideal I of A ( X ) , Z A [ I ] = Z B [ I ∩ B ( X ) ] . We point out an error in the proof of this lemma. The authors have used this lemma to prove a theorem, which says that (a) if M is a maximal ideal of A ( X ) then Z A [ M ] is contained in a unique z-ultrafilter on X and (b) if F is a z-ultrafilter on X, then Z A − 1 [ F ] is a maximal ideal of A ( X ) . The authors have given a correct proof of part (b) of this result, in a more general context, in a later article [Redlin and Watson, 1997]. We give a correct proof of the above lemma and generalize part (a) of the above theorem to prime ideals. Lastly we show that if A ( X ) ≠ C ( X ) , then there exists a non-maximal prime ideal in A ( X ) .