“School of Mathematics”
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Paper IPM / M / 2944  


Abstract:  
The spaces X in which every prime z^{°}ideal of C(X) is
either minimal or maximal are characterized. By this
characterization, it turns out that for a large class of
topological spaces X, such as metric spaces, basically
disconnected spaces and onepoint compactification of discrete
spaces, every prime z^{°}ideal in C(X) is either minimal or
maximal. We will also answer the following questions: When is
every nonregular prime ideal in C(X) a z^{°}ideal? When is
every nonregular (prime) zideal in C(X) a z^{°}ideal? For
instance, we have shown that every nonregular prime ideal of
C(X) is a z^{°}ideal if and only if X is ∂space
(a space in which the boundary of any zeroset is again a zeroset).
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