Flat cover

Any module over a ring has a flat cover that is unique up to (non-unique) isomorphism.

The homomorphism F→M is defined to be a flat cover of M if it is surjective, F is flat, every homomorphism from flat module to M factors through F, and any map from F to F commuting with the map to M is an automorphism of F. While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover.

This flat cover conjecture was explicitly first stated in (Enochs 1981, p 196).

The conjecture turned out to be true, resolved positively and proved simultaneously by Bican, El Bashir & Enochs (2001).

This was preceded by important contributions by P. Eklof, J. Trlifaj and J. Xu.