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.

Key Words:

• QF ring
• Semilocal ring
• The Faith–Menal conjecture

2010 Mathematics Subject Classification:

• 16L60
• 16D50

ACKNOWLEDGMENTS

The author would like to thank the referee for his (her) nice comments. The author would also like to thank Dr. Xiaoxiang Zhang for his nice suggestions.

Notes

Communicated by T. Albu.