# On z◦ -ideals in C(X)

Fundamenta Mathematicae (1999)

• Volume: 160, Issue: 1, page 15-25
• ISSN: 0016-2736

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## Abstract

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An ideal I in a commutative ring R is called a z°-ideal if I consists of zero divisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We characterize topological spaces X for which z-ideals and z°-ideals coincide in , or equivalently, the sum of any two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are characterized in terms of z°-ideals. Finally, we construct two topological almost P-spaces X and Y which are not P-spaces and such that in every prime z°-ideal is either a minimal prime ideal or a maximal ideal and in C(Y) there exists a prime z°-ideal which is neither a minimal prime ideal nor a maximal ideal.

## How to cite

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Azarpanah, F., Karamzadeh, O., and Rezai Aliabad, A.. "On z◦ -ideals in C(X)." Fundamenta Mathematicae 160.1 (1999): 15-25. <http://eudml.org/doc/212377>.

@article{Azarpanah1999,
abstract = {An ideal I in a commutative ring R is called a z°-ideal if I consists of zero divisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We characterize topological spaces X for which z-ideals and z°-ideals coincide in , or equivalently, the sum of any two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are characterized in terms of z°-ideals. Finally, we construct two topological almost P-spaces X and Y which are not P-spaces and such that in every prime z°-ideal is either a minimal prime ideal or a maximal ideal and in C(Y) there exists a prime z°-ideal which is neither a minimal prime ideal nor a maximal ideal.},
author = {Azarpanah, F., Karamzadeh, O., Rezai Aliabad, A.},
journal = {Fundamenta Mathematicae},
language = {eng},
number = {1},
pages = {15-25},
title = {On z◦ -ideals in C(X)},
url = {http://eudml.org/doc/212377},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Azarpanah, F.
AU - Karamzadeh, O.
AU - Rezai Aliabad, A.
TI - On z◦ -ideals in C(X)
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 1
SP - 15
EP - 25
AB - An ideal I in a commutative ring R is called a z°-ideal if I consists of zero divisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We characterize topological spaces X for which z-ideals and z°-ideals coincide in , or equivalently, the sum of any two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are characterized in terms of z°-ideals. Finally, we construct two topological almost P-spaces X and Y which are not P-spaces and such that in every prime z°-ideal is either a minimal prime ideal or a maximal ideal and in C(Y) there exists a prime z°-ideal which is neither a minimal prime ideal nor a maximal ideal.
LA - eng
UR - http://eudml.org/doc/212377
ER -

## References

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8. [8] O. A. S. Karamzadeh, On a question of Matlis, Comm. Algebra 25 (1997), 2717-2726. Zbl0878.16003
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