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Completely Faithful Modules and Self-Injective Rings

Published online by Cambridge University Press:  22 January 2016

Goro Azumaya
Goro Azumaya*
Affiliation:
University of Massachusetts and Indiana University
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A left module over a ring Λ is called completely faithful if Λ is a sum of those left ideals which are homomorphic images of M. The notion was first introduced by Morita [9], and he proved, among others, the following theorem which plays a basic role in his theory of category-isomorphisms: if a Λ-module M is completely faithful, then M is finitely generated and projective with respect to the endomorphism ring Γ of M and Λ coincides with the endomorphism ring of Λ-module M.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 2 , July 1966, pp. 697 - 708
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Auslander, M. and Goldman, O., Maximal orders, Trans. Amer. Math. Soc, 97 (1960), pp. 1–24.Google Scholar
[2] Azumaya, G., Corrections and supplements to my paper concerning Krull-Remak-Schmidt’s theorem, Nagoya Math. J., 1 (1950), pp. 117–124.Google Scholar
[3] Azumaya, G., A duality theory for injective modules, Amer. J. Math., 81 (1959), pp. 249–278.Google Scholar
[4] Bass, H., The Morita theorems, Mimeographed note.Google Scholar
[5] Eckmann, B. and Schopf, A., Über injective Moduln, Arch. Math., 4 (1953), pp. 75–78.Google Scholar
[6] Eilenberg, S. and Nakayama, T., On the dimension of modules and algebras II, Nagoya Math. J., 9 (1955), pp. 1–16.CrossRefGoogle Scholar
[7] Ikeda, M., A characterization of quasi-Frobenius rings, Osaka Math. J., 4 (1952), pp. 203–210.Google Scholar
[8] Jans, J. P., Projective injective modules, Pacific J. Math. 9 (1959), pp. 1103–1108.Google Scholar
[9] Morita, K., Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku, 6 (1958), pp. 83–142.Google Scholar
[10] Morita, K., Category-isomorphisms and endomorphism rings of modules, Trans. Amer. Math. Soc, 103 (1962), pp. 451–469.Google Scholar
[11] Nakayama, T., On Frobeniusean algebras I, Ann. of Math., 40 (1939), pp. 611–633.CrossRefGoogle Scholar
[12] Nakayama, T., On Frobeniusean algebras II, Ann. of Math., 42 (1941), pp. 1–21.Google Scholar
[13] Nakayama, T., On a generalized notion of Galois extensions of a ring, Osaka Math. J., 15 (1963), pp. 11–23.Google Scholar
[14] Nesbitt, C. J. and Thrall, R. M., Some ring theorems with applications to modular representations, Ann. of Math., 47 (1946), pp. 551–567.CrossRefGoogle Scholar