“School of Mathematics”
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| Paper IPM / M / 52 |
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An ideal I in a commutative ring R is called a z°-ideal if I consists of zero-divisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We
characterize topological spaces X for which z-ideals and z°-ideals
coincide in C(X), or equivalently the sum of every two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are
characterized in terms of z°-ideals. Finally, we construct two topological almost P-spaces X and Y which are note P-spaces and such that in C(X) every prime z°-ideal is either a minimal
prime ideal or a maximal ideal and in C(Y) there exists a prime z°-ideal which is neither a minimal prime ideal nor a maximal ideal.
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