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Paper   IPM / M / 491
School of Mathematics
  Title:   On ideals consisting entirely of zero divisors
1.  F. Azarpanah
2.  O. A. S. Karamzadeh
3.  A. Rezai Aliabad
  Status:   Published
  Journal: Comm. Algebra
  Vol.:  28
  Year:  2000
  Pages:   1061-1073
  Supported by:  IPM
Let R be a commutative unitary ring, I be a proper ideal of R and aR. Define Pa as the intersection of all minimal prime ideals containing aI is said to be a z°-ideal if PaI for each aI. Note that such ideals have been studied before under the name of d-ideals. The nilradical of R is the smallest z°-ideal of R. In the study of z°-ideals, it may thus be assumed that R is a reduced ring, by going over to R/rad(R). Different characterizations and examples of z°-ideals are given. The behavior of z°-ideals under extensions and contractions is studied.
The authors show that if R is a reduced ring such that every finitely generated ideal of R consisting of zero divisors has a nonzero annihilator, then any ideal consisting of zero divisors is contained in a z°-ideal.
Necessary and sufficient conditions for the classical ring of quotients of a reduced ring to be regular are given in terms of z°-ideals.

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