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Higher Derivations and Central Simple Algebras

Published online by Cambridge University Press:  22 January 2016

A. Roy  and
R. Sridharan
A. Roy
Affiliation:
Tata Institute of Fundamental Research, Centre for Advanced Study & Research in Mathematics, University of Bombay, BOMBAY
R. Sridharan
Affiliation:
Tata Institute of Fundamental Research, Centre for Advanced Study & Research in Mathematics, University of Bombay, BOMBAY
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Let K be a commutative ring, A a K-algebra, and B a K-subalgebra of A. The object of this paper is to prove some results on higher derivations (in the sense of Jacobson [4]) of B into A. In § 1 we introduce a notion of equivalence among higher derivations. With this notion of equivalence, we prove in § 2 (Theorem 1) that the equivalence classes of higher K-derivations of B into A are in one-one correspondence with the isomorphism classes of certain filtered B ⊗ KA°-modules, where A° denotes the opposite algebra of A.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 21 - 30
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Albert, A.A: Structure of Algebras, Amer. Math. Soc. Colloquium Publications, Vol. 24 (1939).Google Scholar
[2] Hochschild, G: Restricted Lie algebras and simple associative algebras of characteristic p, Trans. Amer. Math. Soc., Vol. 80 (1955), pp. 135–147.Google Scholar
[3] Jacobson, N: Abstract derivation and Lie algebras, Trans. Amer. Math. Soc., Vol. 42 (1937), pp. 206–224.Google Scholar
[4] Jacobson, N: Structure of rings, Amer. Math. Soc. Colloquium Publications, Vol. 37 (1956).Google Scholar