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• Lemma 8 IfA is closed under pure submodules and cokernels of pure monomorphisms, then A is closed
• under pure subcomplexes and cokernels of pure monomorphisms. Proof
• It follows from Lemma 3.7. ✷
• Based on the preceding results, we get the following theorem by analogy with the proof of Theorem 2.8.
• Theorem 3.9 Let (A, B) be a cotorsion pair in R-Mod. If A is closed under pure submodules and cokernels ⊥
• of pure monomorphisms, then the cotorsion pair (A, A) is complete. Furthermore, it is perfect. In [11], Garc´ıa Rozas defined Gorenstein flat complexes and characterized such complexes over Gorenstein
• rings. A complex C is called Gorenstein flat if there exists an exact sequence of complexes· · · −→ F −1−→
• F0−→ F1−→ · · · such that each Fiis flat, C = Ker(F0→ F1) and the sequence remains exact when I⊗−
• is applied to it for any injective complex I . It was proven that every complex over a commutative Gorenstein
• ring has a Gorenstein flat cover [11, Theorem 5.4.8]. We will show that the same result holds if R is a right coherent ring. The following lemma is due to Yang [22, Theorem 5].
• Lemma 10 Let R be a right coherent ring, C a complex. Then C is Gorenstein flat if and only if Cnis
• Gorenstein flat in R -Mod for all n∈ Z. According to the above lemma, it is shown that over a right coherent ring the class of Gorenstein flat
• complexes coincides withGF . Thus we get the following corollary.
• Corollary 3.11 Every complex over a right coherent ring has a Gorenstein flat cover. According to [10, Theorem 2.12], all left modules over a right coherent ring have Gorenstein flat covers.
• Corollary 3.11 shows that the corresponding result holds in the category of complexes of R -modules, and
• generalizes Theorem 5.4.8 in [11]. Analogously, we have the following two corollaries.
• Corollary 3.12 Every complex has aFn-cover.
• Corollary 3.13 Every complex has aMF -cover.