# Flatness criterion: If $K$ is any submodule of a finitely generated free module, then $A \otimes_R K \to A \otimes F$ is injective.

This is question 10.5.25(b) in Dummit and Foote (page 406). I will paste the part of the problem I'm struggling with.

If $A \otimes_R I \to A \otimes_R R$ is injective for every finitely generated ideal $I$, prove that $A \otimes I \to A \otimes_R R$ is injective for every ideal $I$. Show that if $K$ is any submodule of of a finitely generated free module $F$, then $A \otimes_R K \to A \otimes_R F$ is injective.

Another concern I have is the fact that $K$ is an arbitrary submodule. Supposing $F = R$, then we only know that $K$ is a left ideal, since we are not in the setting of commutative rings. So, even if we were to proceed by induction, it is not clear to me how we prove the base case.
• Have you tried inducting on the rank of $F$? If rank is one, it is just the first part. Now, filter the free module and the submodule and see what you get. – Mohan Jan 29 '18 at 15:38