Injective hull of the trivial k[x]-module

Question

Here is the problem

Let k be an algebraically closed field. Describe the injective hull of the trivial $k[x]$-module (The trivial module $k[x]/(x)$).

Someone suggested to me that the answer should be $k[x, x^{-1}]/k[x]$. Why?

Answer 1

Hints: Since $k[x]$ is a principal ideal domain, a module is injective if and only if it is divisible. Now, to show $k[x]$-divisibility of $k[x^{\pm 1}]/k[x]$, note that any element of $k[x]$ has the form $x^n\cdot P$ with $P(0)\neq 0$, so you may reduce to showing that $\cdot x^n$ and $\cdot (1 + xQ)$ are surjective on $k[x^{\pm 1}]/k[x]$. Can you do that?