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

摘要

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 A' 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) is contained in 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 ζ is a z-ultrafilter on X, then Z~(-1)_A [ζ] 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).