= Solution
First show that <free modules are flat>. If $F=\bigoplus_{j\in J}Ae_j$, distributivity of the <tensor product of modules> gives
$$
U\otimes_AF\cong\bigoplus_{j\in J}U.
$$
Explicitly, $u\otimes\sum_j a_je_j$ maps to $(a_ju)_j$, and the inverse sends a finitely supported tuple $(u_j)_j$ to $\sum_j u_j\otimes e_j$. For an <injection> $u:U\hookrightarrow V$, the tensor map is therefore the coordinatewise <injection> $\bigoplus_JU\hookrightarrow\bigoplus_JV$. This proves flatness for any free rank.
By part (c), a <projective module> $P$ is a <direct summand> of a <free module> $F$. The latter is flat by the preceding argument, and part (b) proves that its summand $P$ is <flat>. Therefore $\boxed{\text{every projective }A\text{-module is flat}}$. This proves <projective modules are flat> directly from the exactness and splitting definitions.
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