  Let $G$ be the fundamental group of a three-manifold. By piecing together
many known facts about three manifold groups, we establish two properties of
the group ring $\mathbb{C}G$. We show that if $G$ has rational cohomological
dimension two, then $\mathbb{C}G$ is coherent. We also show that if $G$ is
torsion-free, then $G$ satisfies the Strong Atiyah Conjecture over $\mathbb{C}$
and hence that $\mathbb{C}G$ satisfies Kaplansky's Zero Divisor Conjecture.
