The FGF conjecture and the singular ideal of a ring. (Q2853961)

A ring \(R\) is called left FGF (left CF) if every finitely generated (respectively, cyclic) left \(R\)-module embeds in a free module. The following conjectures, raised by C. Faith in the early 80s, are still open: 1) CF conjecture: Every left CF ring is left Artinian; 2) FGF conjecture: Every left FGF ring is quasi-Frobenius.NEWLINENEWLINE These conjectures have been verified in several special cases, and it is well known that the CF conjecture implies the FGF conjecture. Both conjectures are true if the ring \(R\) has finitely generated essential socle [see \textit{C. Faith}, Advances in non-commutative ring theory, Lect. Notes Math. 951, 21-40 (1982; Zbl 0504.16009)], or if every cyclic left \(R\)-module essentially embeds in a projective module [see \textit{J. L. Gómez Pardo} and \textit{P. A. Guil Asensio}, Trans. Am. Math. Soc. 349, No. 11, 4343-4353 (1997; Zbl 0892.16012)].NEWLINENEWLINE As the main result of this paper, the authors show that both conjectures are true if the ring \(R\) is von Neumann regular modulo its singular left ideal. The structure of left Artinian left CF rings is studied, and it is shown that these rings are left continuous rings.