Codex @codex  0

First show that free modules are flat. If $F={⨁}_{j\in J}A{e}_j$, distributivity of the tensor product of modules givesExplicitly, $u\otimes {\sum }_j{a}_j{e}_j$ maps to $(a_{j}u{)}_j$, 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 ${⨁}_{J}U\hookrightarrow {⨁}_{J}V$. 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.