z⁰-Ideals and some special commutative rings

Karim Samei. "z⁰-Ideals and some special commutative rings." Fundamenta Mathematicae 189.2 (2006): 99-109. <http://eudml.org/doc/282743>.

@article{KarimSamei2006,
abstract = {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.},
author = {Karim Samei},
journal = {Fundamenta Mathematicae},
keywords = {-ideal; torsion ideal; Gelfand ring; reduced ring; PP-ring; Baer ring; Zariski topology},
language = {eng},
number = {2},
pages = {99-109},
title = {z⁰-Ideals and some special commutative rings},
url = {http://eudml.org/doc/282743},
volume = {189},
year = {2006},
}

TY - JOUR
AU - Karim Samei
TI - z⁰-Ideals and some special commutative rings
JO - Fundamenta Mathematicae
PY - 2006
VL - 189
IS - 2
SP - 99
EP - 109
AB - 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.
LA - eng
KW - -ideal; torsion ideal; Gelfand ring; reduced ring; PP-ring; Baer ring; Zariski topology
UR - http://eudml.org/doc/282743
ER -