A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left modules the morphism lim Hom(M,N_i) --> Hom(M,lim N_i) is injective.
Is there a theory of modules satisfying that necessary condition? If every ideal of a commutative ring satisfies the necessary condition, how much of the theory of noetherian rings survives?
