  Let $G$ be an abelian group of order $n$ and let $R$ be a commutative ring
which admits a homomorphism ${\Bbb Z}[\zeta_{n}]\ra R$, where $\zeta_{n}$ is a
(complex) primitive $n$-th root of unity. Given a finite $R[G\e]$-module $M$,
we derive a formula relating the order of $M$ to the product of the orders of
the various isotypic components $M^{\e\chi}$ of $M$, where $\chi$ ranges over
the group of $R$-valued characters of $G$. We then give conditions under which
the order of $M$ is exactly equal to the product of the orders of the
$M^{\chi}$. To derive these conditions, we build on work of E.Aljadeff and
obtain, as a by-product of our considerations, a new criterion for
cohomological triviality which improves the well-known criterion of T.Nakayama.
We also give applications to abelian varieties and to class groups of abelian
fields, obtaining in particular some new class number formulas. Our results
also have applications to "non-semisimple" Iwasawa theory, but we do not
develop these here. In general, the results of this paper can be used to
strengthen a variety of known results involving finite $R[G\e]$-modules whose
hypotheses include (an equivalent form of) the following assumption: ``the
order of $G$ is invertible in $R$".}
