Covers by flat modules and submodules of flat modules (Q1119718)

Let C be a class of left R-modules, where R is a unitary ring. One says that a left R-module has a cover f: \(T\to E\) in C if the following two conditions are fulfilled: (i) \(T\in C\), f is an R-linear map such that whenever \(f': T'\to E\) is another R-linear map with \(T'\in C\), then there is a linear map (over R) u: T\({}'\to T\) such that \(f\circ u=f'\), and (ii) if v: \(T\to T\) is an R-linear map such that \(f\circ v=f\) then v is an automorphism of T. The aim of this paper is to prove the following results: 1) If R is a right coherent ring (e.g. if R is commutative and noetherian) then all left R-modules have covers by submodules iff all injective R-modules have flat covers. 2) If R is commutative, noetherian of finite Krull dimension, then all cotorsion R-modules have flat covers. In this case, if E is a flat R-module then E[[x]] is a flat R[[x]]- module.