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For a ring $R$, a left $R$-module $F$ is flat if the functor $- \otimes_R F$ brings an exact sequence of right $R$-modules to an exact sequence of right $R$-modules.

It is faithfully flat if the functor $- \otimes_R F$ brings a sequence of right $R$-modules to an exact sequence of right $R$-modules if and only if the original sequence is exact.

It is a "standard fact" that the sum of a faithfully flat module and a flat module is faithfully flat. Can someone please explain why this is true.

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The tensored sequence splits as two exact sequences, and it suffices to use the exactness of one to recover the exactness of the untensored sequence.

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  • $\begingroup$ Yes, of course! Thanks for your help. $\endgroup$ – Dyke Acland Aug 18 '16 at 15:04

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