Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T15:14:30.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Semi-Prime Rings Whose Homomorphic Images are Serial

Published online by Cambridge University Press:  20 November 2018

Lawrence S. Levy  and
Patrick F. Smith
Lawrence S. Levy
Affiliation:
University of Wisconsin, Madison, Wisconsin
Patrick F. Smith
Affiliation:
University of Glasgow, Glasgow, Scotland
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.

A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.

We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 691 - 695
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Cartan, H. and Eilenberg, S., Homologuai algebra (Princeton, 1956.Google Scholar
2. Faith, C., On Kôethe rings, Math. Annalen 164 (1966), 207–212.Google Scholar
3. Faith, C., Algebra II ring theory (Springer-Verlag, 1976.Google Scholar
4. Goldie, A. W., Semiprime rings with maximum condition, Proc. London Math. Soc. 10 (1960), 201–220.Google Scholar
5. Jategaonkar, A. V., A counter-example in ring theory and homological algebra, J. Algebra 12 (1969), 418–440.Google Scholar
6. Klatt, G. B. and Levy, L. S., Pre-self-injective rings, Trans. Amer. Math. Soc. 137 (1969), 407–419.Google Scholar
7. Levy, L. S., Torsion-free and divisible modules over non-integral-domains, Can. J. Math. 15 (1963), 132–151.Google Scholar
8. Levy, L. S., Commutative rings whose homomorphic images are self-injective, Pacific J. Math. 18 (1966), 149–153.Google Scholar
9. Matlis, E., Injective modules over Prufer rings, Nagoya Math. J. 15 (1959), 57–59.Google Scholar
10. Sharpe, D. W. and Vamos, P., Injective modules (Cambridge University Press, 1972.Google Scholar
11. Singh, S., Modules over hereditary Noetherian prime rings, Can. J. Math. 27 (1975), 867–883.Google Scholar
12. Smith, P. F., Rings with every proper image a principal ideal ring, Proc. Amer. Math. Soc. (to appear).Google Scholar
13. Warfield, R. B., Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), 187–222.Google Scholar
14. Zaks, A., Some rings are hereditary, Israel J. Math. 10 (1971), 442–450.Google Scholar