infinite indeterminates over a left noetherian ring is left coherent(want a basic proof)

Question

A ring is left coherent if every finitely generated left ideal is finitely presented.

Statement: If $R$ is a left noetherian ring, then $R[X]$ is left coherent where $X$ represents infinite many indeterminates. I do not see any basic proof without using exactness of $lim$ functor (Polynomial ring in infinitely many variables over a noetherian ring is coherent).

This is a statement made in Rotman, Homological algebra chapt 3. It has not yet reached the direct limit functor part. So I guess he assures some basic proof without using even limit functor.

Answer 1

If a left ideal $J$ in $R[X]$ is finitely generated, one can choose its finite set of generators in the subring $R[Y]$ for some finite subset $Y$ of $X$. Then the module of relations $\Omega$ of $J$ (with respect to the chosen set of generators) is generated by the same elements as the module of relations of the ideal $J'$ in $R[Y]$ generated by the same elements as $J$. The last module is finitely generated since the ring $R[Y]$ is Noetherian.

(Of course, one can consider this argument as a shadow of some more general proof based on the properties of limit.)