Basis for Module Over PID

Question

Given $M$ is a free module over PID (let's say R). $$X = \{x_1, x_2, \cdots, x_m\}$$ span $M$. Can $X$ can be reduced, so that $X$ is a basis for $M$?

Note: $X$ is a basis for module-$M$ if $X$ span $M$ and $X$ linearly independent.

Answer 1

In general, the answer is no. Consider $M=\mathbb{Z}$ as a (free) $\mathbb{Z}$-module, and $X = \{2,3\}$. By Bézout's Identity, $X$ spans $\mathbb{Z}$, but neither $\{2\}$ nor $\{3\}$ do.