Abstract A ring R is called left FGF (resp. CF) if every finitely generated (resp. cyclic) left R-module embeds in a free module. In this paper we give new partial affirmative answers to two open problems in the area of Module and Ring Theory, the so-called FGF problem: Is a left FGF ring Quasi-Frobenius? and the CF problem: Is a left CF ring left artinian?. ∗The second author has been partially supported by the D.G.E.S. of Spain (PB96-0961-C02-02) and the Fundación ”Senec ” of the Comunidad Autonomade Murcia (expediente nurnero 00260/CU/97) ∗The second author has been partially supported by the D.G.E.S. of Spain (PB96-0961-C02-02) and the Fundación ”Senec ” of the Comunidad Autonomade Murcia (expediente nurnero 00260/CU/97) Notes ∗The second author has been partially supported by the D.G.E.S. of Spain (PB96-0961-C02-02) and the Fundación ”Senec ” of the Comunidad Autonomade Murcia (expediente nurnero 00260/CU/97)