Let $I, J$ be ideals of a ring $R$ and assume $I$ is flat as a right $R$-module.
I would like to show that Showing $I \otimes J \cong IJ$. 

Here is my strategy: Let $j : J \rightarrow R$ be the inclusion map. Then by flatness $I \otimes j$ is injective. I would then like to conclude that $Im (I \otimes j) = IJ$ and thus complete the proof, but I am uncertain about this step - is it true that $Im (I \otimes j) = IJ$? It seems correct, but I am having trouble formulating why this should be true.