Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 3 / b / Solution

An $R$-module $F$ is flat when $-{\otimes }_{R}F$ preserves injections, equivalently all finite exact sequences. Since $-{\otimes }_{R}R$ 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,is free. Therefore the polynomial algebra $R[X]$ is a flat $R$-algebra.