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Rank Element of a Projective Module

Published online by Cambridge University Press:  22 January 2016

Akira Hattori
Akira Hattori*
Affiliation:
Tokyo University of Education
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In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 25 , 1965, pp. 113 - 120
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

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