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We introduce an invariant, called mean rank, for any module M of the integralgroup ring of a discrete amenable group $\Gamma$, as an analogue of the rank ofan 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 vonNeumann-L\"uck rank of M all coincide.  As applications, we establish an addition formula for mean dimension ofalgebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem foralgebraic actions, and show that for elementary amenable groups with an upperbound on the orders of finite subgroups, algebraic actions with zero meandimension are inverse limits of finite entropy actions.

Mean Dimension, Mean Rank, and von Neumann-L\"uck Rank