may think of group rings most of the time. However, crossed products show up naturally, e.g., in proving that the virtual Ore dimension  is well defined.

We recall the following definition.

\[def:goldie\_ring\] A ring $R$ is *left Goldie* if there exists $d\in \mathbb{N}$ such that every direct sum of nonzero left ideals of $R$ has at most $d$ summands and the left annihilators $a(x)=\{r\in R;~rx=0\}$, $x\in R$, satisfy the maximum condition for ascending chains. A ring $R$ is *prime* if for any two ideals $A,B$ in $R$, $AB=0$ implies $A=0$ or $B=0$.

The subgroup of $G$ generated by its finite normal subgroups will be denoted by $\Delta^+(G)$. Then $\Delta^+(G)$ is also the set of elements of finite order which have only finitely many conjugates. We need the following three results:

\[lem:delta\_implies\_prime\] If $\Delta^+(G) = 1$, then $k*G$ is prime.

\[thm:prime\_Goldie\_matrix\] The set of non-zero-divisors in a prime left Goldie ring satisfies the Ore condition. The Ore localization $S^{-1}R$ is isomorphic to $\operatorname{M}_d(D)$ for some $d \in \mathbb{N}$ and some skew field $D$.

\[thm:elementary\_amenable\_goldie\] If $G$ is amenable and $k*G$ is a domain, then $k*G$ is a prime left Goldie ring. If $G$ is an elementary amenable group such that the 