  We establish a technique to prove that a Koszul graded ring is prime or a
domain using information about its Koszul dual. This is based on a general
categorical result that expands on methods of J.Y. Guo, which proves that
certain orbital rings are prime or domains. We apply this method to prove that
if $A = kQ/I$ is a Koszul twisted Calabi-Yau algebra of dimension 2, such that
$Q$ is connected with every vertex having outdegree at least 2, then $A$ is a
prime piecewise domain. In particular, the preprojective algebra of a connected
quiver whose underlying graph has minimum degree at least 2 is a prime
piecewise domain.
