  Let $R$ be a commutative ring, $\pi$ be a finite group, $R\pi$ be the group
ring of $\pi$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and
$\pi$ is a finite group. Then the Cartan map $c:K_0(R\pi)\to G_0(R\pi)$ is
injective. Theorem 2. Suppose that $R$ is a Dedekind domain with
$\fn{char}R=p>0$ and $\pi$ is a $p$-group. Then every finitely generated
projective $R\pi$-module is isomorphic to $F \oplus \c{A}$ where $F$ is a free
module and $\c{A}$ is a projective ideal of $R\pi$. Moreover, $R$ is a
principal ideal domain if and only if every finitely generated projective
$R\pi$-module is isomorphic to a free module. Theorem 3. Let $R$ be a
commutative noetherian ring with total quotient ring $K$, $A$ be an $R$-algebra
which is a finitely generated $R$-projective module. Suppose that $I$ is an
ideal of $R$ such that $R/I$ is artinian. Let $\{\c{M}_1,\ldots,\c{M}_n\}$ be
the set of all maximal ideals of $R$ containing $I$. Assume that the Cartan map
$c_i: K_0(A/\c{M}_iA)\to G_0(A/\c{M}_iA)$ is injective for all $1\le i\le n$.
If $P$ and $Q$ are finitely generated $A$-projective modules with $KP\simeq
KQ$, then $P/IP\simeq Q/IQ$.
