The purpose of this
paper is to characterize quasi-Frobenius, QF, rings in terms of relationships
assumed to exist for each cyclic or finitely generated left module between
the module and its second dual, where duality is with respect to the ring.
More specifically we prove that a left perfect ring is QF if every cyclic left
module is reflexive or every finitely generated left module is (isomorphic to) a
submodule of a free module. For rings with minimum condition on left or
right ideals this later condition is equivalent to every finitely generated left
module being torsionless or to the ring being a cogenerator in the category of
finitely generated left modules. If annihilator relations are defined by means of
the natural pairing between a module and its dual, this condition is also
equivalent to every submodule of a finitely generated left module being an
annulet.
