On $z^\circ$-ideals in $C(X)$

Tom 160 / 1999

F. Azarpanah, O. A. S. Karamzadeh, A. Rezai Aliabad Fundamenta Mathematicae 160 (1999), 15-25 DOI: 10.4064/fm_1999_160_1_1_15_25


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 , or equivalently, the sum of any 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 not P-spaces and such that in 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.


  • F. Azarpanah
  • O. A. S. Karamzadeh
  • A. Rezai Aliabad

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