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.

Any module M over a ring has a resolution by flat modules

→ F₂ → F₁ → F₀ → M → 0

such that each F_n+1 is the flat cover of the kernel of F_n → F_n−1. Such a resolution is unique up to isomorphism, and is a minimal flat resolution in the sense that any flat resolution of M factors through it. Any homomorphism of modules extends to a homomorphism between the corresponding flat resolutions, though this extension is in general not unique.

• Enochs, Edgar E. (1981), "Injective and flat covers, envelopes and resolvents", Israel Journal of Mathematics, 39 (3): 189–209, doi:10.1007/BF02760849, ISSN 0021-2172, MR 0636889
• Bican, L.; El Bashir, R.; Enochs, E. (2001), "All modules have flat covers", Bulletin of the London Mathematical Society, 33 (4): 385–390, doi:10.1017/S0024609301008104, ISSN 0024-6093, MR 1832549
• "Flat cover", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Xu, Jinzhong (1996), Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Berlin: Springer-Verlag, doi:10.1007/BFb0094173, ISBN 3-540-61640-3, MR 1438789