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On Absolutely Segregated Algebras and Relative 3-Cohomology Groups of an Algebra

Published online by Cambridge University Press:  22 January 2016

Tadasi Nakayama
Tadasi Nakayama*
Affiliation:
Mathematical Institute, Nagoya University
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Recently M. Ikeda [1] succeeded in determining the structure of absolutely segregated algebras, i.e. algebras whose 2-cohomology groups all vanish. His beautiful result reads : an algebra A, of finite rank over its ground field, is absolutely segregated if and only if i) the residue-algebra A/N modulo the radical N is separable and, moreover, ii) the A-left-module N is an (Mo)-module. A. simplification was given by H. Nagao [5], who obtained, besides an interesting-result on algebras with vanishing 3-(or higher) cohomology groups, an elegant short proof to the fact that under the assumption of i), the property ii) is necessary, and sufficient, for the absolute segregation of A.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 6 , October 1953 , pp. 177 - 185
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

[1] Hochschild, G., On the cohomology groups of an associative algebra, Ann. Math. 46 (1945), 58–67.Google Scholar
[2] Hochschild, G., On the cohomology theory for associative algebras, Ann. Math. 47(1646), 568–579.Google Scholar
[3] Hochschild, G., Cohomology and representations of associative algebras, Duke Math. J. 14 (1947), 921–948.CrossRefGoogle Scholar
[4] Ikeda, M., On absolutely segregated algebras, Nagoya Math. J., this number.Google Scholar
[5] Nagao, H., Note on the cohomology groups of associative algebras, Nagoya Math. J., this number.Google Scholar
[6] Nagao, H.-Nakayama, T., On the structure of (M0 )- and (Mu )-modules, forthcoming in Math. Zeitschr.Google Scholar