Sofic Mean Length

Given a length function L on the R-modules of a unital ring R, for each sofic group $\mathrm{\Gamma }$ we define a mean length for every locally L-finite $R\mathrm{\Gamma }$-module relative to a bigger $R\mathrm{\Gamma }$-module. We establish an addition formula for the mean length. We give two applications. The first one shows that for any unital left Noetherian ring R, $R\mathrm{\Gamma }$ is stably direct finite. The second one shows that for any $Z\mathrm{\Gamma }$-module M, the mean topological dimension of the induced $\mathrm{\Gamma }$-action on the Pontryagin dual of M coincides with the von Neumann-L"{u}ck rank of M.