In a commutative ring R, an ideal I consisting entirely of zero divisors is called a torsion ideal, and an ideal is called a z⁰-ideal if I is torsion and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We prove that in large classes of rings, say R, the following results hold: every z-ideal is a z⁰-ideal if and only if every element of R is either a zero divisor or a unit, if and only if every maximal ideal in R (in general, every prime z-ideal) is a z⁰-ideal, if and only if every torsion z-ideal is a z⁰-ideal and if and only if the sum of any two torsion ideals is either a torsion ideal or R. We give a necessary and sufficient condition for every prime z⁰-ideal to be either minimal or maximal. We show that in a large class of rings, the sum of two z⁰-ideals is either a z⁰-ideal or R and we also give equivalent conditions for R to be a PP-ring or a Baer ring.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-1,
author = {Karim Samei},
title = {z0-Ideals and some special commutative rings},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {99-109},
zbl = {1101.13002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-1}
}
Karim Samei. z⁰-Ideals and some special commutative rings. Fundamenta Mathematicae, Tome 189 (2006) pp. 99-109. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-1/