Let $R$ be a commutative DVR, and let $M$ be the free $R$-module of finite rank $k\ge 2$. Let $N$ be a submodule of $M$ isomorphic to $R$.
Is it true that $N$ is a direct summand of $M$?
Thanks in advance.
In most examples, R has principal ideals I whose corresponding quotients are torsion modules. it follows that I is not a summand of R, and that $I\oplus 0$ is not a summand of $R^2$.