Abstract
The Hecke-Kiselman algebra of a finite oriented graph Θ over a field K is studied. If Θ 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.
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