Geodesic
Semi-Prime Rings Whose Homomorphic Images are Serial
Canadian journal of mathematics, Tome 34 (1982) no. 3, pp. 691-695

Voir la notice de l'article provenant de la source Cambridge University Press

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”. 
Levy, Lawrence S.; Smith, Patrick F. Semi-Prime Rings Whose Homomorphic Images are Serial. Canadian journal of mathematics, Tome 34 (1982) no. 3, pp. 691-695. doi: 10.4153/CJM-1982-046-4
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[1] 1. Cartan, H. and Eilenberg, S., Homologuai algebra (Princeton, 1956. Google Scholar

[2] 2. Faith, C., On Kôethe rings, Math. Annalen 164 (1966), 207–212. Google Scholar

[3] 3. Faith, C., Algebra II ring theory (Springer-Verlag, 1976. Google Scholar

[4] 4. Goldie, A. W., Semiprime rings with maximum condition, Proc. London Math. Soc. 10 (1960), 201–220. Google Scholar

[5] 5. Jategaonkar, A. V., A counter-example in ring theory and homological algebra, J. Algebra 12 (1969), 418–440. Google Scholar

[6] 6. Klatt, G. B. and Levy, L. S., Pre-self-injective rings, Trans. Amer. Math. Soc. 137 (1969), 407–419. Google Scholar

[7] 7. Levy, L. S., Torsion-free and divisible modules over non-integral-domains, Can. J. Math. 15 (1963), 132–151. Google Scholar

[8] 8. Levy, L. S., Commutative rings whose homomorphic images are self-injective, Pacific J. Math. 18 (1966), 149–153. Google Scholar

[9] 9. Matlis, E., Injective modules over Prufer rings, Nagoya Math. J. 15 (1959), 57–59. Google Scholar

[10] 10. Sharpe, D. W. and Vamos, P., Injective modules (Cambridge University Press, 1972. Google Scholar

[11] 11. Singh, S., Modules over hereditary Noetherian prime rings, Can. J. Math. 27 (1975), 867–883. Google Scholar

[12] 12. Smith, P. F., Rings with every proper image a principal ideal ring, Proc. Amer. Math. Soc. (to appear). Google Scholar

[13] 13. Warfield, R. B., Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), 187–222. Google Scholar

[14] 14. Zaks, A., Some rings are hereditary, Israel J. Math. 10 (1971), 442–450. Google Scholar

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