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Algebras With Vanishing n-Cohomology Groups

Published online by Cambridge University Press:  22 January 2016

Masatoshi Ikeda ,
Hiroshi Nagao  and
Tadashi Nakayama
Masatoshi Ikeda
Affiliation:
Osaka University, Osaka City University, Nagoya University
Hiroshi Nagao
Affiliation:
Osaka University, Osaka City University, Nagoya University
Tadashi Nakayama
Affiliation:
Osaka University, Osaka City University, Nagoya University
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Cohomology theory for (associative) algebras was first established in general higher dimensionalities by G. Hochschild [3], [4], [5]. Algebras with vanishing 1-cohomology groups are separable semisimple algebras ([3], Theorem 4.1). On extending and refining our recent results [6], [8], [12], we establish in the present paper the following:

Let n ≧ 2. Let A be an (associative) algebra (of finite rank) possessing a unit element 1 over a field Ω, and N be its radical.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 7 , June 1954 , pp. 115 - 131
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1954

References

[ 1 ] Brauer, R.-Nesbitt, C., On the regular representations of algebras, Proc. Nat. Acad. Sci. 23 (1937), 236–0.CrossRefGoogle ScholarPubMed
[ 2 ] Cartan, H.-Eilenberg, S., Homological Algebra, Princeton University Press, forthcoming.Google Scholar
[ 3 ] Hochschild, G., On the cohomology groups of an associative algebra, Ann. Math. 46 (1945), 58–7.Google Scholar
[ 4 ] Hochschild, G., On the cohomology theory for associative algebras, Ann. Math. 47 (1946), 568–9.CrossRefGoogle Scholar
[ 5 ] Hochschild, G., Cohomology and representations of associative algebras, Duke Math. J. 14 (1947), 921–8.Google Scholar
[ 6 ] Ikeda, M., Cn absolutely segregated algebras, Nagoya Math. J. 6 (1953), 63–5.Google Scholar
[ 7 ] Jacobson, N., Theory of Rings, New York, 1943.CrossRefGoogle Scholar
[ 8 ] Nagao, H., Note on the cohomology groups of associative algebras, Nagoya Math. J. 6 (1953), 85–2.Google Scholar
[ 9 ] Nagao, H.-Nakayama, T., On the structure of (Mo )- and (Mu )-modules, Math. Zeitschr. 59(1953), 164–0.CrossRefGoogle Scholar
[10] Nakayama, T., Some studies on regular representations, induced representations and modular representations, Ann. Math. 39 (1938), 361–9.Google Scholar
[11] Nakayama, T., Derivations and cohomology in simple and other rings, I, Duke Math. J. 19 (1952), 51–3.CrossRefGoogle Scholar
[12] Nakayama, T., On absolutely segregated algebras and relative 3-cohomology groups of an algebra, Nagoya Math. J. 6 (1953), 177–5.Google Scholar
[13] Rose, I. H., On the cohomology theory for associative algebras, Amer. J. Math. 74 (1952), 531–6.Google Scholar
[14] Shih, K. S., Dissertation Univ. of Illinois, 1953.Google Scholar