Radicals of incidence algebras

Abstract

The incidence algebra of a locally finite partially ordered set X; with the partial ordering "≤", over a ring with identity T is defined as the set of all mappings f : X x X ---T where f(x; y) = 0 for all x; y 2 X with x 6· y and denoted by I(X; T): The operations on I(X; T) are given by (f + g)(x; y) = f(x; y) + g(x; y) (f ¢ g)(x; y) = X x·z·y f(x; z) ¢ g(z; y) (r ¢ f)(x; y) = rf(x; y) for f; g 2 I(X; T); r 2 T and x; y 2 X: When the ring R is commutative, the ring I(X;R) becomes an algebra. The aim of this study is to investigate some special radicals of incidence algebras and determine the necessary and sufficient conditions characterizing elements of these radicals by using the very definition of the strong product property.