A Generalization of Π-coherent Rings

A ring R is said to be a right AF-ring if the right annihilator of every left ideal is finitely generated. In this paper, some equivalent characterizations and properties of AF-rings are listed. We prove that if R is left self P-injective and right AF-ring, then R is a right coherent ring. As a consequence, we prove that if R is an AF-ring, then R is an IF-ring if and only if R is self FP-injective. Finally, we consider AF-rings and their significance for the duality theory. In particular, it is proved that R is a QF ring if and only if R is left perfect ring and R is self FP-injective, under the condition that R is a right AF-ring.