New characterizations of Artinian modules and hereditary rings (Q1177846)

Let \(R\) be an associative ring with 1. The author proves that a left \(R\)- module \(M\) is Artinian iff every non-empty set of finitely cogenerated factor modules of \(M\) has a maximal element, and that \(R\) is left hereditary iff in every projective left \(R\)-module \(M\) the intersection \(U\cap V\) of two direct summands \(U\) and \(V\) of \(M\) is also a direct summand of \(M\).