On the maximal spectrum of commutative semiprimitive rings

K. Samei

Colloquium Mathematicae (2000)

  • Volume: 83, Issue: 1, page 5-13
  • ISSN: 0010-1354

Abstract

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The space of maximal ideals is studied on semiprimitive rings and reduced rings, and the relation between topological properties of Max(R) and algebric properties of the ring R are investigated. The socle of semiprimitive rings is characterized homologically, and it is shown that the socle is a direct sum of its localizations with respect to isolated maximal ideals. We observe that the Goldie dimension of a semiprimitive ring R is equal to the Suslin number of Max(R).

How to cite

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Samei, K.. "On the maximal spectrum of commutative semiprimitive rings." Colloquium Mathematicae 83.1 (2000): 5-13. <http://eudml.org/doc/210774>.

@article{Samei2000,
abstract = {The space of maximal ideals is studied on semiprimitive rings and reduced rings, and the relation between topological properties of Max(R) and algebric properties of the ring R are investigated. The socle of semiprimitive rings is characterized homologically, and it is shown that the socle is a direct sum of its localizations with respect to isolated maximal ideals. We observe that the Goldie dimension of a semiprimitive ring R is equal to the Suslin number of Max(R).},
author = {Samei, K.},
journal = {Colloquium Mathematicae},
keywords = {spectrum of a ring; maximal ideal space},
language = {eng},
number = {1},
pages = {5-13},
title = {On the maximal spectrum of commutative semiprimitive rings},
url = {http://eudml.org/doc/210774},
volume = {83},
year = {2000},
}

TY - JOUR
AU - Samei, K.
TI - On the maximal spectrum of commutative semiprimitive rings
JO - Colloquium Mathematicae
PY - 2000
VL - 83
IS - 1
SP - 5
EP - 13
AB - The space of maximal ideals is studied on semiprimitive rings and reduced rings, and the relation between topological properties of Max(R) and algebric properties of the ring R are investigated. The socle of semiprimitive rings is characterized homologically, and it is shown that the socle is a direct sum of its localizations with respect to isolated maximal ideals. We observe that the Goldie dimension of a semiprimitive ring R is equal to the Suslin number of Max(R).
LA - eng
KW - spectrum of a ring; maximal ideal space
UR - http://eudml.org/doc/210774
ER -

References

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  1. [1] M. Contessa, On PM-rings, Comm. Algebra 10 (1982), 93-108. 
  2. [2] G. De Marco and A. Orsatti, Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Amer. Math. Soc. 30 (1971), 459-466. Zbl0207.05001
  3. [3] R. Engelking, General Topology, PWN-Polish Sci. Publ., 1977. 
  4. [4] L. Gillman, Rings with Hausdorff structure space, Fund. Math. 45 (1957), 1-16. Zbl0079.26301
  5. [5] 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
  6. [6] O. A. S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184. Zbl0524.54013
  7. [7] C. W. Kohls, The space of prime ideals of a ring, Fund. Math. 45 (1957), 17-27. Zbl0079.26302
  8. [8] G. Mason, z-ideals and prime ideals, J. Algebra 26 (1973), 280-297. Zbl0262.13003
  9. [9] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley Interscience, New York, 1987. Zbl0644.16008
  10. [10] S. H. Sun, Noncommutative rings in which every prime ideal is contained in a unique maximal ideal, J. Pure Appl. Algebra 76 (1991), 179-192. Zbl0747.16001
  11. [11] S. H. Sun, Rings in which every prime ideal is contained in a unique maximal ideal, ibid. 78 (1992), 183-194. Zbl0774.16001

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