of rank two as subgroup, then the Ore condition is never satisfied for $kG$ [@Linnell(2006)]\*[Proposition 2.2]{}.

From the previous remark and the discussion above we obtain:

\[the:Ore-dimension\] Let $G$ be a torsionfree amenable group such that $kG$ contains no zero-divisors. Then the Ore dimension $\dim^{{\operatorname{Ore}}}_{kG}$ has the following properties:

1.  $\dim^{{\operatorname{Ore}}}_{kG}(kG) = 1$;

2.  For any short exact sequence of $kG$-modules $0 \to M_0 \to M_1 \to
        M_2\to 0$ we get $$\dim^{{\operatorname{Ore}}}_{kG}(M_1) = \dim^{{\operatorname{Ore}}}_{kG}(M_0) + \dim^{{\operatorname{Ore}}}_{kG}(M_2).$$

Crossed products, Goldie rings, and the generalized Ore localization 
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Throughout, let $G$ be a group, let $k$ be a skew field.

Let $R$ be a ring. The notion of crossed product generalizes the one of group ring. A *crossed product* $R\ast G=R\ast_{c,\tau} G$ is determined by maps $c:G\to{\operatorname{aut}}(R)$ and $\tau:G\times G\to R^\times$ such that, roughly speaking, $c$ is a homomorphism up to the $2$-cocycle $\tau$. We refer to the survey [@Lueck(2002)]\*[10.3.2 on p. 398]{} for details. If $G$ is an extension of $H$ by $Q$, then the group ring $kG$ is isomorphic to a crossed product $kH\ast Q$. Some results in this paper are formulated for crossed products, although we only need the case of group rings for Theorem \[the:dim\_approximation\_over\_fields\]. So the reader 