The flat cover conjecture, that is, ‘every module has a flat cover’, has been formulated formally in 1981 by E. E. Enochs [Isr. J. Math. 39, 189-209 (1981; Zbl 0464.16019)], though it was implicitly raised in 1959 by H. Bass [Trans. Am. Math. Soc. 95, 466-488 (1960; Zbl 0094.02201)].
This paper contains not one, but two different proofs of the flat cover conjecture. The first of these proofs (due to E. Enochs) uses the concept of a cotorsion theory and relies on work done by P. C. Eklof and J. Trlifaj [Bull. Lond. Math. Soc. 33, No. 1, 41-51 (2001; Zbl 1030.16004)].
The second proof (due to L. Bican and R. El Bashir) is more direct. For a class of modules \(\mathcal A\), the purity generated by \(\mathcal A\) is defined as the class of all short exact sequences with respect to which all \(A\in\mathcal A\) are projective. For a purity \(\sigma\), a module \(F\) is called \(\sigma\)-coprojective if any exact sequence \(0\to N\to M\to F\to 0\) belongs to \(\sigma\). It is shown that for \(\sigma\) a purity projectively generated by a set of finitely presented modules over any ring, every module has a \(\sigma\)-coprojective cover.
Since the classical purity is generated by (all) finitely presented modules, and pure coprojectve modules are exactly the flat modules, the conjecture is proved.