Let $R$ be an integral domain such that every finitely generated ideal of $R$ is principal. Show that every finitely generated submodule of $R^n$ is free.
We know if $R$ is a PID, then the statement is true, but how about over a Bezout domain? Any help is appreciated.
Hint: Let $M$ be a finitely generated submodule of $R^n$, $I_1:=\{r\in R\mid\text{ there exists }s\in R^{n-1}\text{ such that } (r,s)\in M\}$, $M_{1}:=\{s\in R^{n-1}\mid(0,s)\in M\}$. Prove that $I_1$ and $M_1$ are finitely generated (and consequently, $I_1$ is principal) and $M\simeq I_1\oplus M_1$, then induct on $n$.