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Paper   IPM / M / 8013
School of Mathematics
  Title:   z-ideals and √{z° }-ideals in C(X)
  Author(s):  F. Azarpanah (Joint with R. Mohamadian)
  Status:   Published
  Journal: Acta Math. Sin. (Engl. Ser.)
  Vol.:  23
  Year:  2007
  Pages:   989-996
  Supported by:  IPM
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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