I am looking for a counterexample to a simple question about proper sub-modules. The book I am reading mentions the following theorem but implys that there are pathological examples related to the theorem when one considers non-commutative rings.
Let $D$ be a principal ideal domain, let $n\in \mathbb{Z}$ and let $D^{(n)}$ denote a free $D$-module of rank $n$.
Theorem: If $L$ is a submodule of $D^{(n)}$ then $L$ is a free $D$-module of rank $m \leq n$
Question: If $L$ is proper submodule of $D^{(n)}$ must the rank of $L$ satisfy $m < n$ ?
