Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-17T14:55:41.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Subrings of the Maximal Ring of Quotients Associated With Closure Operations

Published online by Cambridge University Press:  20 November 2018

D. C. Murdoch
D. C. Murdoch*
Affiliation:
The University of British Columbia, Vancouver, British Columbia
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 contains a number of results that grew out of an attempt to solve the following problem : Given a non-commutative ring R with suitable ascending chain condition, and a prime ideal P in R, to construct a corresponding local ring RP in which the extension P′ of P is a unique maximal prime, and to prove, if possible, that the intersection of the powers of P′ is zero. The present investigation is at best a preliminary attack on this problem since the contribution to the complete solution is comparatively small and the central problem of the intersection of the powers of P′ has not been touched.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 723 - 743
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Barnes, W. E., Primal ideals and isolated components in noncommutative rings, Trans. Amer. Math. Soc, 82 (1956), 1–16.Google Scholar
2. Findlay, G. D. and J. Lambek, A generalized ring of quotients II, Can. Math. Bull., 1 (1958), 155–167.Google Scholar
3. Goldie, A. W., Rings with maximum condition (Yale University, 1961, mimeographed notes).Google Scholar
4. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895.Google Scholar
5. Johnson, R. E., Structure theory of faithful rings I. Closure operations on lattices, Trans. Amer. Math. Soc, 84 (1957), 508–522.Google Scholar
6. Johnson, R. E., Equivalence rings, Duke Math. J. 15 (1948), 787–793.Google Scholar
7. Lesieur, L. and Croisot, R., Théorie Noethérienne des anneaux, des demi-groupes et des modules dans le cas noncommutatif, II, Math. Ann., 134 (1958), 458–476.Google Scholar
8. McCoy, Neal H., Prime ideals in general rings, Amer. J. Math., 71 (1949), 823–833.Google Scholar
9. Murdoch, D. C., Contributions to noncommutative ideal theory, Can. J. Math., 4 (1952), 43–57.Google Scholar
10. Querré, Julien, Equivalences de fermeture dans un demi-groupe residutif, Sem. Dubreil-Pisot (Algèbre et théorie des nombres), 15e année, 1961-62, No. 3.Google Scholar
11. Utumi, Y., On quotient rings, Osaka Math. J., 8 (1956), 1–18.Google Scholar
12. Zariski, O. and Samuel, P., Commutative algebra, Vol. 1 (Princeton, N.J., 1958).Google Scholar