  We introduce an invariant, called mean rank, for any module M of the integral
group ring of a discrete amenable group $\Gamma$, as an analogue of the rank of
an abelian group. It is shown that the mean dimension of the induced
$\Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von
Neumann-L\"uck rank of M all coincide.
  As applications, we establish an addition formula for mean dimension of
algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for
algebraic actions, and show that for elementary amenable groups with an upper
bound on the orders of finite subgroups, algebraic actions with zero mean
dimension are inverse limits of finite entropy actions.
