if and only if it hold for $UA$. This follows from , $\ker U = 0$, $\dim ({\operatorname{coker}}U) =
\dim_{k*G}^{{\operatorname{Ore}}} ({\operatorname{coker}}U) = 0$, and in the second case we use the third property of $\dim$.

We may write $S^{-1}k*G = \operatorname{M}_d(D)$ for some $d\in \mathbb{N}$ and some skew field $D$. By applying the Morita equivalence from $\operatorname{M}_d(D)$ to $D$ and back and doing Gaussian elimination over $D$ we see that there are invertible matrices $U,V \in \operatorname{M}_{rd}(S^{-1}k*G)$ such that $U\operatorname{diag}(A,\dots,A)V = J$, where there are $d$ $A$’s and $J$ is a matrix of the form $\operatorname{diag}(1,\dots,1,0,\dots,0)$. Now choose $u,v \in S$ such that $uU,vV \in
\operatorname{M}_{rd}(k*G)$. Then $(uU) \operatorname{diag}(A,\dots,A) (Vv) = uJv$, and the result follows.

\[the:Comparing\_Eleks\_dimension\_and\_the\_Ore\_dimension\] Let $G$ be a finitely generated group and let $k$ be a skew field. Suppose that $kG$ is a prime left Goldie ring. Then for any finitely presented left $kG$-module $M$ $$\dim^{{\operatorname{Elek}}}_{kG} (M) = \dim_{kG}^{{\operatorname{Ore}}}(M).$$

This follows from Theorem \[the:Comparing\_dimensions\] and Theorem \[the:Eleks-dimension\].

Proof of the main theorem {#sec:Proof_of_the_main_theorem}
=========================

 \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] In the first step we reduce the claim to the case, where $G$ is finitely generated. Consider a finitely presented left $kG$-module $M$. Choose a matrix $A \in M_{r,s}(kG)$ such that $M$ 