 Free modules are flat

For a free module $F={⨁}_{i\in J}A{e}_i$, the natural map $U{\otimes }_{A}F\to {⨁}_{i\in J}U$ sends $u\otimes {\sum }_i{a}_i{e}_i$ to $(a_{i}u{)}_i$ 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 ${⨁}_{J}U\hookrightarrow {⨁}_{J}V$. Thus every free module is flat, including those of infinite rank.