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On the Dimension of Modules and Algebras IX: Direct Limits

Published online by Cambridge University Press:  22 January 2016

Israel Berstein
Israel Berstein*
Affiliation:
Bucharest, Roumania
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Let J be a directed set and let {Λj, φij} be a direct system of rings indexed by J and with limit Λ. Let {Aj, φij} be a direct system of groups indexed by J. Assume that each Aj is a left Λj-module and that φij(λa) = φij(λ) φij(a) for each λ ∈ Λj, a ∈ Aj. Then the limit A of (Aj, φij) is a left Λ-module.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 13 , June 1958 , pp. 83 - 84
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1958

References

[1] Kuročkin, V. M., The decomposition of algebras into a semi-direct sum of the radical and a semi-simple subalgebra, C. R. (Doklady) Acad. Sci., URSS (N.S.) vol. 36 (1942), pp. 42–45.Google Scholar
[2] Rosenberg, A. and Zelinsky, D., Cohomology of infinite algebras, Trans. Amer. Math. Soc., vol. 82 (1956), pp. 85–98.Google Scholar