When $M$ is a submodule of $M[t^{-1}]$?

Question

Take $X=\mathbb{C}[[t]]$ and $M$ an $X$ module. I was reading the following statement:

If M is flat as $X-$module then $M$ can be seen as submodule of $M[t^{-1}]$

Now, I think that I can prove it using the flatness, the idea is watching $$ M \cong M\otimes_XX \hookrightarrow M \otimes \mathbb{C}[[t]][t^{-1}]\cong M[t^{-1}].$$

Now, I think that is always true that $M$ is an $X$-submodule of $M[t^{-1}]$. I'd appreciate any type of help

Answer 1

The map $M \to M[t^{-1}], m \mapsto \frac{m}{1},$ is injective if and only if the map $M \to M, m \mapsto tm,$ is.