ABSTRACT
One of the main goals in the ring theory is to reduce in a certain sense the description of large classes of rings to simpler classes using some ring theoretic constructions. The most classical example is the Wedderburn-Artin theorem, however one can mention a number of other general results. This chapter represents the definition and main properties of a basic construction of rings. These are called incidence rings of posets over associative rings. The chapter considers some properties of a special class of incidence rings of the form T(S) = I(S, D), where S is a finite partially ordered set and D is a division ring. This class of rings properly contains the class of hereditary serial rings and all Artinian rings with quivers that are trees. The chapter introduce and study a special class of incidence rings. These are called incidence rings modulo the radical.
