Let $M$ be a (not necessarily finitely generated) free module over a PID $R$. Let $X$ be its basis and let $Y$ be a proper subset of $X$. Let $N$ be a submodule of $R$ and let $N_Y = \langle Y \rangle\cap N$. If $N_Y$ is free with a basis $B$, can we conclude that $|B| \leq |Y|$?
Grillet (the textbook Abstract Algebra) implicitly uses this result in his proof that every submodule of a free module $M$ over PID is free with the dimension less or equal to that of $M$.
Hint: comsider $K$ the field of fraction of $R$. denote $M_K$ the $K$-vector space associated to $M$ (the localization of $M$). $dim_KM_K=dim_RM$, since the result is true for $K$, it is true for $R$.