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.
This question was asked here.
However, I am looking for a more elementary proof that does not involve colimits or the Tor functor. For the first part of the question above, I believe it is easily solved just by looking at the construction of the tensor product. I'm having particular trouble with the second part.
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.