Ideal approximation theory

Let (A;E) be an exact category and F⊆Ext a subfunctor. A morphism φ in A is an F-phantom if the pullback of an E-conflation along φ is a conflation in F. If the exact category (A;E) has enough injective objects and projective morphisms, it is proved that an ideal I of A is special precovering if and only if there is a subfunctor F⊆Ext with enough injective morphisms such that I is the ideal of F-phantom morphisms. A crucial step in the proof is a generalization of Salce's Lemma for ideal cotorsion pairs: if I is a special precovering ideal, then the ideal cotorsion pair (I,I⊥) generated by I in (A;E) is complete. This theorem is used to verify: (1) that the ideal cotorsion pair cogenerated by the pure-injective modules of R-Mod is complete; (2) that the ideal cotorsion pair cogenerated by the contractible complexes in the category of complexes Ch(R-Mod) is complete; and, using Auslander and Reiten's theory of almost split sequences, (3) that the ideal cotorsion pair cogenerated by the Jacobson radical Jac(Λ-mod) of the category Λ-mod of finitely generated representations of an Artin algebra is complete.