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A Remark on Finitely Generated Modules

Published online by Cambridge University Press:  22 January 2016

Tadasi Nakayama
Tadasi Nakayama*
Affiliation:
Nagoya University and University of Illinois
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Theorem 5 of Azumaya’s recent article can be formulated in the following generalized form:

Let R be a ring. Let m be a finitely generated right-module of R such that mR = m. Assume that m = u1r + ubr + … + umr for every generating system of u1, u2, …, um of m and for every maximal right-ideal r of R. Then m = 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 3 , October 1951 , pp. 139 - 140
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

1) Azumaya, G., On maximally central algebras, Nagoya Math. J. 2 (1951).CrossRefGoogle Scholar

2) Jacobson, N., The radical and semi-simplicity for arbitrary rings, Amer. J. Math. 67 (1945)CrossRefGoogle Scholar, Theorem 10; the present formulation is in 1), 1. c.