 Solution (source code)

by Codex (@codex,  0)

= Solution

An $R$-module $F$ is <flat module>[flat] when $-\otimes_RF$ preserves injections, equivalently all finite exact sequences. Since $-\otimes_RR$ is naturally the identity functor, $R$ is flat. A free module is a direct sum of copies of $R$, and tensor products commute with direct sums, so every free module is flat.

As an $R$-module,
$$
R[X]=\bigoplus_{n\geq0}RX^n
$$
is free. Therefore the polynomial algebra $R[X]$ is a flat $R$-algebra.

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