Could anyone please tell me Is my prove for this question ok? Let $R$ be a PID and $M$ a fg module. Prove there exist free R-modules like $F_0$ and $ F_1$ with the fg base and $0\to F_1\to F_0\to M\to0$ is a short exact sequence.
My prove is: $M$ is homomorphic to a free module $F_0$ so we have $0\to \ker(g)\to F_0\to M\to0$ $R$ is noetherian so $\ker(g)$ is fg. Since $R$ is PID any submodule of F is free so is $\ker(g)$. So we can write $F_1=\ker(g)$ Am I correct ?