Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T04:58:56.006Z Has data issue: false hasContentIssue false

On the Ring of Quotients of a Noetherian Ring

Published online by Cambridge University Press:  20 November 2018

J. Lambek*
Affiliation:
McGill University and Summer Research InstituteCanadian Mathematical Congress
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.

This paper is largely an expository account of known facts, but it contains at least one result believed to be new, Proposition 6.

Our main technique is the method of lifting idempotents developed in Part I. This has been treated in the literature, but not quite in the generality required here. It turns out that much of classical artinian ring theory can be done for the semi-perfect rings introduced by Bass, as will have been noticed by many other people.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Bass, H., Finitistic homological dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488.CrossRefGoogle Scholar
2. Eckmann, B. and Schopf, A., Űber Injektive Moduln, Archiv der Math. 4 (1953), 75-78.CrossRefGoogle Scholar
3. Faith, C., Lectures on infective modules and quotient rings, Rutgers 1964. (Multilithed.)Google Scholar
4. Feller, E. H. and Swokowski, E. W., On ring extensions for completely primary noncommutative rings, Trans. Amer. Math. Soc. 105 (1962), 251-263.CrossRefGoogle Scholar
5. Findlay, G.D. and Lambek, J., A generalized ring of quotients II, Can. Math. Bull. 1 (1958), 155-167.CrossRefGoogle Scholar
6. Goldie, A. W., The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. 8 (1958), 589-608.CrossRefGoogle Scholar
7. Goldie, A. W., Semi-prime rings with maximum condition, Proc. London Math. Soc. 10 (1960), 201-220.CrossRefGoogle Scholar
8. Goldie, A. W., Rings with maximum condition, Yale 1961. (Multilithed.)Google Scholar
9. Jacobson, N., Structure of rings, Providence 1956.CrossRefGoogle Scholar
10. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895.CrossRefGoogle Scholar
11. Johnson, R. E., Structure theory of faithful rings, II. Restricted rings, Trans. Amer. Math. Soc. 84 (1957), 523-544. III. Irreducible rings, Proc. Amer. Math. Soc.11 (1960), 710-717.Google Scholar
12. Johnson, R. E., Quotient rings of rings with zero singular ideal, Pac. J. Math. 11 (1961), 1385-1392.CrossRefGoogle Scholar
13. Johnson, R. E. and Wong, E. T., Quasi-injective modules and irreducible rings, J. London Math. Soc. 36 (1961), 260-268.CrossRefGoogle Scholar
14. Lambek, J., On Utumi' s ring of quotients, Can. J. Math. 15 (1963), 363-370.CrossRefGoogle Scholar
15. Lesieur, L. and Croisot, R., Algèbre noethérienne non commutative, Paris 1963.Google Scholar
16. Lesieur, L. and Croisot, R., Coeur d' un module, Journal de Mathématique 42 (1963), 367-407.Google Scholar
17. Matlis, E., Injective modules over noetherian rings, Pac. J. Math. 8 (1958), 511-528.CrossRefGoogle Scholar
18. Talentyre, T. D., Quotient rings of rings with maximum condition on right ideals, J. London Math. Soc. 38 (1963), 439-450.CrossRefGoogle Scholar
19. Utumi, Y., On quotient rings, Osaka Math. J. 8 (1956), 1-18.Google Scholar
20. Utumi, Y., On a theorem on modular lattices, Proc. Japan Acad. 35 (1959), 16-21.Google Scholar