People often say that a module $M$ over a not necessarily neotherian local ring $R$ being projective is flat, and also free. However, some refer to the finitely-generatedness of $M$, i.e. $M$ being generated by $m_1, \ldots, m_n$ with $n < \infty$ over $R$. Then I would like to ask that when ${\mathrm{Spec}}\,R[X] \to {\mathrm{Spec}}\,R$ is flat, as it is, the affine ring $R[X]$ is flat over $R$. In this case, am I to think that the free decomposition $$ R[X] = R \oplus RX\oplus RX^2 \ldots, $$ which is necessarily of infinite length?
I have never seen any reference referring to this infiniteness. To the best of my knowledge, $R[X]$ is flat over any ring $R$, though.