Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T21:26:18.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Localization in Non-Commutative Noetherian Rings

Published online by Cambridge University Press:  20 November 2018

Bruno J. Müller
Bruno J. Müller*
Affiliation:
McMaster University, Hamilton, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

To construct a well behaved localization of a noetherian ring R at a semiprime ideal S, it seems necessary to assume that the set (S) of modulo S regular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ring Rs in addition: one calls such 5 classical. To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 600 - 610
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.Google Scholar
2. Bass, H., Finitistic dimension and a homologuai generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
3. Borho, W., Gabriel, P. und Rentschler, R., Primideale in Einhùllenden auflosbarer Lie- Algebren, Lecture Notes in Mathematics 357 (Springer 1973).Google Scholar
4. Chatters, A. W. and Ginn, S. M., Localization in hereditary rings, J. Algebra 22 (1972), 82–88.Google Scholar
5. Chatters, A. W. and Heinicke, A. G., Localization at a torsion theory in hereditary noetherian rings, Proc. London Math. Soc. 27 (1973), 193–204.Google Scholar
6. Eisenbud, D., Subrings of artinian and noetherian rings, Ann. Math. 195 (1970), 247–249.Google Scholar
7. Eisenbud, D. and Robson, J. C., Hereditary noetherian prime rings, J. Algebra 16 (1970), 86–104.Google Scholar
8. Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.Google Scholar
9. Goldie, A. W., The structure of noetherian rings, Lecture Notes in Mathematics 246 (Springer 1972), 213–321.Google Scholar
10. Huppert, B., Endliche Gruppen I (Springer 1967).Google Scholar
11. Jategaonkar, A. V., Infective modules and classical localization in noetherian rings, Bull. Amer. Math. Soc. 79 (1973), 152–157.Google Scholar
12. Jategaonkar, A. V., The torsion theory at a semiprime ideal, A. Acad. Brasil. Cienc. 5 (1973), 197–200.Google Scholar
13. Jategaonkar, A. V., Injective modules and localization in non-commutative noetherian rings, Trans. Amer. Math. Soc. 188 (1974), 109–123.Google Scholar
14. Kuzmanovich, J., Localization in HNP rings, Trans. Amer. Math. Soc. 173 (1972), 137–157.Google Scholar
15. Lambek, J. and Michler, G. O., The torsion theory at a prime ideal of a right-no ether ian ring, J. Algebra 25 (1973), 364–389.Google Scholar
16. Lambek, J. and Michler, G. O., Completions and classical localizations of right noetherian rings, Pacific J. Math. J+8 (1973), 133–140.Google Scholar
17. Lambek, J. and Michler, G. O., Localization of right noetherian rings at semiprime ideals, Canad. J. Math. 26 (1974), 1069–1085.Google Scholar
18. Lenagan, T. H., Bounded hereditary noetherian prime rings, J. London Math. Soc. 6 (1973), 241–246.Google Scholar
19. Ludgate, A. T., A. note on non-commutative noetherian rings, J. London Math. Soc. 5 (1972), 406–408.Google Scholar
20. McConnell, J. C., Localization in enveloping rings, J. London Math. Soc. J+3 (1968), 421- 428; erratum and addendum, ibid. 3 (1971), 409–410.Google Scholar
21. Michler, G. O., Primringe mit Krull-Dimension Eins, J. Reine Angew. Math. 239-2%0 (1970), 366–381.Google Scholar
22. Michler, G. O., The kernel of a block of a group algebra, Proc. Amer. Math. Soc. 37 (1973), 47–49.Google Scholar
23. Michler, G. O., The blocks of p-nilpotent groups over arbitrary fields, J. Algebra 24- (1973), 303–315.Google Scholar
24. Small, L. W., Orders in artinian rings, J. Algebra 4 (1966), 13–41.Google Scholar
25. Smith, P. F., Localization and the Artin-Rees property, Proc. London Math. Soc. 22 (1971), 39–68.Google Scholar
26. Smith, P. F., Localization in group rings, Proc. London Math. Soc. 22 (1971), 69–90.Google Scholar
27. Stenstrôm, B., Rings and modules of quotients, Lecture Notes in Mathematics 237 (Springer 1971).Google Scholar
28. Swan, R. G., K-theory of finite groups and orders, Lecture Notes in Mathematics 149 (Springer 1970).Google Scholar
29. Zariski, O. and Samuel, P., Commutative algebra II (Van Nostrand 1960).Google Scholar