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Azumaya’s Canonical Module and Completions of Algebras

Published online by Cambridge University Press:  22 January 2016

James Osterburg
James Osterburg*
Affiliation:
University of Cincinnati, Taft Fellow and Indiana University
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We are concerned with an algebra S over a commutative ring. Precisely S is a non-commutative ring with identity which is also a finitely generated unital R module such that r(xy) = (rx)y = x(ry) for r in R and x, y ∈ S. In section one, we assume A is a commutative, Artinian ring. Following Goro Azumaya (see (1, p. 273)), we define the canonical module F of A to be the injective hull of A modulo the Jacobson radical of A i.e. F = I(A/J(A)).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 49 , March 1973 , pp. 9 - 19
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

1) Azumaya, Goro, “A Duality Theory for Injective Modules (Theory for Quasi Frobenius Modules)”, American J. of Math. 81 (1959), pp. 249–278.Google Scholar
2) Azumaya, Goro, “On Maximally Central Algebras”, Nagoya Math. J., 2 (1951), pp.119–150.Google Scholar
3) Cartan, H. and Eilenberg, S., Homological Algebra, Princeton, N. J.: Princeton University Press, 1956.Google Scholar
4) Matlis, Eben, “Injective Modules over Noetherian Rings”, Pacific J. of Math., 8 (1958), pp. 511–528.Google Scholar
5) Sandomierski, F. L., “Some Examples of Right Self Injective Rings which are not left Self Injective”, P.A.M.S., 28 (1970), pp. 244–245.Google Scholar