A Note on the Faith–Menal Conjecture

Abstract

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.