“School of Mathematics”
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| Paper IPM / M / 491 |
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| Abstract: | |||||||
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Let R be a commutative unitary ring, I be a proper ideal of
R and a ∈ R. Define Pa as the intersection of all minimal
prime ideals containing a. I is said to be a z°-ideal
if Pa ⊂ I for each a ∈ I. 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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