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Generalized Casimir Operators

Published online by Cambridge University Press:  20 November 2018

A. J. Douglas
A. J. Douglas*
Affiliation:
The University, Sheffield, England
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Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.

In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1496 - 1505
Copyright
Copyright © Canadian Mathematical Society 1969

References

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