orders of the finite subgroups are bounded, then $k*G$ is left Goldie.

If $G$ is amenable and $k\ast G$ is a domain, then $k\ast G$ satisfies the Ore condition [@Dodziuk-Linnell-Mathai-Schick_Yates(2003)]\*[Theorem 6.3]{}, thus its Ore localization with respect to $S=k\ast G-\{0\}$ is a skew field. By [@Passman(1977)]\*[Theorem 4.10 on p. 456]{} $k\ast G$ is a prime left Goldie ring. The second assertion is taken from [@Kropholler-Linnell-Moody(1988)]\*[Proposition 4.2]{}.

Next we extend the definition of Ore dimension to prime left Goldie rings. Let $R$ be such a ring. The functor $S^{-1}R \otimes_R -$ will still be exact [@Stenstroem(1975)]\*[Proposition II.1.4 on page 51]{}. If $M$ is a left $R$-module, then $S^{-1}R \otimes_R M$ will be a direct sum of $n$ irreducible $S^{-1}R$-modules for some non-negative integer $n$, and then the *(generalized) Ore dimension* of $M$ is defined as $$\dim_R^{{\operatorname{Ore}}}(M) = \frac{n}{d}.$$ Since $S^{-1}R\cong\operatorname{M}_d(D)$ (Theorem \[thm:prime\_Goldie\_matrix\]) and $\operatorname{M}_d(D)$ decomposes into $d$ copies of the irreducible module $D^d$, we have $\dim_R^{{\operatorname{Ore}}}(R)=1$.

Elek’s dimension function {#sec:Eleks_dimension_function}
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Throughout this subsection let $G$ be a finitely generated amenable group. We review Elek’s definition [@Elek(2003c)] of a dimension function $\dim^{{\operatorname{Elek}}}_{kG}$ for finitely generated $kG$-modules.

Fix a finite set of generators and equip $G$ with the associated word metric $d_G$. 