“School of Mathematics”
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| Paper IPM / M / 8013 |
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| Abstract: | |
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It is well-known that every prime ideal minimal over a z-ideal is
also a z-ideal. The converse is also well-known in C(X). Thus
whenever I is an ideal in C(X), then √I is a z-ideal 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 z-ideal (z°-ideal) in C(X) which
are not in a chain is a prime z-ideal (z°-ideal). We also
show that every decomposable z-ideal (z°-ideal) in C(X)
is the intersection of a finite number of prime z-ideals
(z°-ideal). Some counter examples in general rings and
some characterizations for the largest (smallest) z-ideal and
z°-ideal contained in (containing) an ideal are given.
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