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Let A be a right R-module. Suppose for every left ideal J of R, the homomorphism $f:A\otimes J\to A$ defined by $f(x\otimes y)=xy$ is injective, then A is a flat R-module.(the identity 1 is in R)

I found the result in my book but didn't give me the proof, and I felt very confused about it! Can someone tell me how to prove it?

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  • $\begingroup$ Does this theorem assume that $1 \in R$? Because if so, then $f$ is just the inclusion $J \to R$ tensored by $A$. That feels like it would be important in a proof, and it shows why the given condition is even related to flatness as defined by "the functor $A\otimes -$ preserves exactness". $\endgroup$ – Arthur Mar 7 '16 at 8:03
  • $\begingroup$ Yes, the identity 1 is in R, $\endgroup$ – python3 Mar 7 '16 at 14:54
  • $\begingroup$ In addition to the linked duplicate, also see this or this that showed up when you used the search feature for your question. You did use the search feature before posting your question, right? $\endgroup$ – rschwieb Mar 7 '16 at 15:00