In [@Elek(2006strong)] Elek proved  for amenable $G$ and $k={{\mathbb C}}$ (see also [@Pape(2008)]).

Moreover, if $G$ is a torsionfree amenable group such that ${{\mathbb C}}G$ contains no zero-divisors and $k$ is a field of characteristic zero, then $$\dim_{{{\mathcal N}}(G)}\bigl({{\mathcal N}}(G) \otimes_{kG} M\bigr) = \dim^{{\operatorname{Ore}}}_{kG}(M).$$ This follows from [@Lueck(2002) Theorem 6.37 on page 259, Theorem 8.29 on page 330, Lemma 10.16 on page 376, and Lemma 10.39 on page 388]. In particular, Theorem \[the:dim\_approximation\_over\_fields\] follows for $k$ of characteristic zero. So the interesting new case is the one of a field of prime characteristic.

Review of Ore localization and Elek’s dimension function {#sec:Review_of_Ore_localization_and_Eleks_dimension_function}
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Ore localization {#sub:ore_localization}
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We review the Ore localization of rings. For proofs and more information the reader is referred to [@Stenstroem(1975)]. Consider a torsionfree group $G$ and a field $k$. Let $S$ be the set of non-zero-divisors of $k G$. This is a multiplicatively closed subset of $k G$ and contains the unit element of $k G$. Suppose that $k G$ satisfies the *Kaplansky Conjecture* or *zero-divisor conjecture*, i.e., $S=k G-\{0\}$. Further assume that $S$ satisfies the *left Ore condition*, i.e., for $r \in k G$ and $s \in S$ there exists $r' \in k G$ 