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Paper   IPM / M / 52
School of Mathematics
  Title:   On z°-ideals in C(X)
1.  A. Rezai Aliabad
2.  F. Azarpanah
3.  O. A. S. Karamzadeh
  Status:   Published
  Journal: Fund. Math.
  Vol.:  160
  Year:  1999
  Pages:   15-25
  Supported by:  IPM
An ideal I in a commutative ring R is called a z°-ideal if I consists of zero-divisors and for each aI 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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