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Given a non-commutative ring R, how to prove that a projective left module is a flat left module by using the natural isomorphism: $Hom_{\mathbb{Z}}(A\otimes_RB,G)\cong Hom_R(B,Hom_{\mathbb{Z}}(A,G))$, where A and B are right module and left module respectively and G is an Abelian group?

Karoo Yang
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    Do you need to use the tensor-hom adjunction? It really just seems simpler to show that direct sum of modules is flat if and only if each piece is flat, and then use the fact that every projective is a direct summand of a free (and free modules are obviously flat). – Alex Youcis Oct 31 '13 at 08:09
  • @AlexYoucis Yes you're right, I need to use tensor-hom adjunction. The method you mentioned seems really much simpler. Thx :) – Karoo Yang Oct 31 '13 at 08:19

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