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Paper IPM / M / 491  


Abstract:  
Let R be a commutative unitary ring, I be a proper ideal of
R and a ∈ R. Define P_{a} as the intersection of all minimal
prime ideals containing a. I is said to be a z^{°}ideal
if P_{a} ⊂ I for each a ∈ I. Note that such ideals have
been studied before under the name of dideals. 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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