ON COHERENCE OF ENDOMORPHISM RINGS

Let R be a ring and U a left R-module with S = End(U-R). The aim of this paper is to characterize when S is coherent. We first show that a left R-module F is T-U-flat if and only if Hom(R) (U, F) is a flat left S-module. This removes the unnecessary hypothesis that U is Sigma-quasiprojective from Proposition 2.7 of Gomez Pardo and Hernandez ['Coherence of endomorphism rings', Arch. Math. (Basel) 48(1) (1987), 40-52]. Then it is shown that S is a right coherent ring if and only if all direct products of T-U-flat left R-modules are T-U-flat if and only if all direct products of copies of U-R are T-U-flat. Finally, we prove that every left R-module is T-U-flat if and only if S is right coherent with wD(S) <= 2 and U-S is FP-injective.