isomorphism. This proves .

\[the:Extension\_to\_the\_virtually\_torsionfree\_case\] Let $G$ be an amenable group which possesses a subgroup $E \subseteq G$ of finite index such that $kE$ is left Goldie and $\Delta^+(E)=1$, and let $k$ be a skew field. Then assertions \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\], \[the:dim\_approximation\_over\_fields:chain\_complexes\] and \[the:dim\_approximation\_over\_fields:CW-complexes\] of Theorem \[the:dim\_approximation\_over\_fields\] remain true, provided we replace $\dim_{kG}^{{\operatorname{Ore}}}$ by ${\operatorname{vdim}}_{kG}^{{\operatorname{Ore}}}$ everywhere.

It suffices to prove the claim for assertion \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] since the proof in Theorem \[the:dim\_approximation\_over\_fields\] that it implies the other two assertions applies also in this more general situation. Let $(G_n)_{n\ge 0}$ be a residual chain of $G$. To prove the result in general, we may assume that $G$ is finitely generated. Since $kE$ is left Goldie and $[G:E]<\infty$, the group ring $kG$ is also left Goldie. Further, every $kG_n$ is left Goldie. Since $\Delta^+(G)$ is finite (its order is bounded by $[G:E]$), there exists $N \in\mathbb{N}$ such that $G_N \cap \Delta^+(G) = 1$, and then $\Delta^+(G_N) = 1$ and $G_i \subseteq G_N$ for all $i \ge N$. Set $H = G_N$, so that $kH$ is prime by Lemma \[lem:delta\_implies\_prime\]. Then for a finitely presented $kH$-module $L$, $$\dim_{kH}^{{\operatorname{Ore}}}(L) = 
\lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{k[G_n\cap H]}
L\bigr)}{[H:H \cap G_n]}$$ by Theorems \[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\] and \[the:Comparing\_dimensions\]. We have $[G:G_n] 