$M \hookrightarrow M[t^{-1}]$

Question

Take $M$ an $X=\mathbb{C}[[t]]$-module.

first question I've to prove that: If $M$ is flat then $M$ is a submodule of $M[t^{-1}]$.

Here my attempt.

We have: $$M\cong M\otimes_X\mathbb{C}[[t]] \hookrightarrow M \otimes\mathbb{C}((t)) \cong M[t^{-1}],$$ where the isomorphisms are intuitive ones and the immersion is given by flatness of $M$ and $\mathbb{C}((t))=\mathrm{frac}(\mathbb{C}[[t]])$.

Is this right?

second question I would like to know a counterexample of the precedent statement in the case with $M$ not flat.

Answer 1

If $M=\mathbb C[[t]]/(t)$, then $M[t^{-1}]=0$, so $M$ is not a submodule of $M[t^{-1}]$.