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On the Structure of Semi-Prime Rings and their Rings of Quotients

Published online by Cambridge University Press:  20 November 2018

Joachim Lambek
Joachim Lambek*
Affiliation:
Institute for Advanced Study and McGill University
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We are mainly interested in the study of prime and semi-prime rings and their rings of quotients. However, our argument proceeds largely in the category of modules (§ 1 to 4) and bimodules (§ 5 to 7).

After a brief description of the generalized rings of quotients introduced recently by Johnson, Utumi, and Findlay and the present author, we study a closure operation on the lattice of submodules of a module. For the lattice of left ideals of a ring, the concept of closed submodules reduces to the If-ideals of Utumi. The lattice of closed submodules of a module is always a complete modular lattice. We are specially interested in the case when it is a complemented lattice. This happens, in particular, when the singular submodule of Johnson and Wong vanishes. We consider the lattice of closed right ideals of a prime ring S and determine the maximal ring of right quotients of S in the case when this lattice has atoms.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 392 - 417
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Birkhoff, G., Lattice theory (New York, 1948).Google Scholar
2. Brainerd, B. and Lambek, J., On the ring of quotients of a Boolean ring, Can. Math. Bull., 2 (1959), 25–29.Google Scholar
3. Dieudonné, J., Les idéaux minimaux dans les anneaux associatifs, Proc. Inter. Congr. Math., vol. II (1950), 44–48.Google Scholar
4. Eckmann, B. and Schopf, A., Über injektive Moduln, Arch, der Math., 4 (1956), 75–78.Google Scholar
5. Findlay, G. D. and Lambek, J., A generalized ring of quotients I, II, Can. Math. Bull., 1 (1958), 77–85 155-167.Google Scholar
6. Gillman, L. and Jerison, M., Rings of continuous functions (New York, 1960).Google Scholar
7. Goldie, A. W., Decompositions of semi-simple rings, J. London Math. Soc, 31 (1956), 40–48.Google Scholar
8. Goldie, A. W., The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. (3), 8 (1958), 589–608.Google Scholar
9. Jacobson, N., Structure of rings (Providence, 1956).Google Scholar
10. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc, 2 (1951), 891–895.Google Scholar
11. Johnson, R. E., Semi-prime rings, Trans. Amer. Math. Soc, 76 (1954), 375–388.Google Scholar
12. Johnson, R. E., Structure theory of faithful rings I, II, Trans. Amer. Math. Soc, 84 (1957), 508- 522, 523–544.Google Scholar
13. Halmos, P. R., Boolean algebra (mimeographed, Chicago, 1959).Google Scholar
14. Kaplanski, I., Infinite abelian groups (Ann Arbor, 1954).Google Scholar
15. Lesieur, L. and Croisot, R., Anneaux premiers Noethériens à gauche, Ann. Sci. Ec. Norm. Sup. (3), 76 (1959), 161–183.Google Scholar
16. McCoy, N. H., Prime ideals in general rings, Amer. J. Math., 71 (1949), 823–833.Google Scholar
17. Utumi, Y., On quotient rings, Osaka Math. J., 8 (1956), 1–18.Google Scholar
18. Utumi, Y., On a theorem on modular lattices, Proc. Japan Acad., 25 (1959), 16–21.Google Scholar
19. Wong, E. T. and Johnson, R. E., Self-injective rings, Can. Math. Bull., 2 (1959), 167–173.Google Scholar