  The Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over a field
$K$ is studied. If $\Theta$ is an oriented cycle, it is shown that the algebra
is semiprime and its central localization is a finite direct product of matrix
algebras over the field of rational functions $K(x)$. More generally, the
radical is described in the case of PI-algebras, and it is shown that it comes
from an explicitly described congruence on the underlying Hecke-Kiselman
monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman
algebra and it is a finite module over its center.
