= Free modules are flat
{title2=$U\otimes_AA^{(J)}\cong U^{(J)}$}
For a <free module> $F=\bigoplus_{i\in J}Ae_i$, the natural map $U\otimes_AF\to\bigoplus_{i\in J}U$ sends $u\otimes\sum_i a_ie_i$ to $(a_iu)_i$ and is an <isomorphism>, with inverse sending the tuple $(u_i)$ to $\sum_i u_i\otimes e_i$. For any <injection> $U\hookrightarrow V$, its <tensor product of modules> with $F$ is therefore the coordinatewise <injection> $\bigoplus_JU\hookrightarrow\bigoplus_JV$. Thus every <free module> is flat, including those of infinite rank.
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