On z◦ -ideals in C(X)

F. Azarpanah; O. Karamzadeh; A. Rezai Aliabad

Fundamenta Mathematicae (1999)

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

Abstract

top
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

top

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

top
  1. [1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addi-son-Wesley, Reading, Mass., 1969. Zbl0175.03601
  2. [2] F. Azarpanah, Essential ideals in , Period. Math. Hungar. 3 (1995), no. 12, 105-112. 
  3. [3] F. Azarpanah, Intersection of essential ideals in , Proc. Amer. Math. Soc. 125 (1997), 2149-2154. Zbl0867.54023
  4. [4] G. De Marco, On the countably generated z-ideals of , ibid. 31 (1972), 574-576. Zbl0211.25902
  5. [5] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976. 
  6. [6] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130. Zbl0147.29105
  7. [7] C. B. Huijsmans and B. de Pagter, On z-ideals and d-ideals in Riesz spaces. I, Indag. Math. 42 (Proc. Netherl. Acad. Sci. A 83) (1980), 183-195. 
  8. [8] O. A. S. Karamzadeh, On a question of Matlis, Comm. Algebra 25 (1997), 2717-2726. Zbl0878.16003
  9. [9] O. A. S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of , Proc. Amer. Math. Soc. 93 (1985), 179-184. Zbl0524.54013
  10. [10] R. Levy, Almost P-spaces, Canad. J. Math. 2 (1977), 284-288. Zbl0342.54032
  11. [11] A. I. Veksler, p'-points, p'-sets, p'-spaces. A new class of order-continuous measures and functions, Soviet Math. Dokl. 14 (1973), 1445-1450. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.