“School of Mathematics”
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Paper IPM / M / 8013  


Abstract:  
It is wellknown that every prime ideal minimal over a zideal is
also a zideal. The converse is also wellknown in C(X). Thus
whenever I is an ideal in C(X), then √I is a zideal if
and only if I is, in which case √I=I. We show the same
fact for z^{°}ideals and then it turns out that the sum of
a primary ideal and a zideal (z^{°}ideal) in C(X) which
are not in a chain is a prime zideal (z^{°}ideal). We also
show that every decomposable zideal (z^{°}ideal) in C(X)
is the intersection of a finite number of prime zideals
(z^{°}ideal). Some counter examples in general rings and
some characterizations for the largest (smallest) zideal and
z^{°}ideal contained in (containing) an ideal are given.
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