Rank Element of a Projective Module

Abstract

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 P 2 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]).