Isomorphic $\mathbb{C}[X]$-modules My question is :

It is true that $\mathbb{C}[X]/(x-c)$ and $\mathbb{C}[X]/(x-d)$ are isomorphic $\mathbb{C}[X]$-modules if and only if $c=d$?

I have the feeling that the answer is simple but I haven't found it.

Let $f:\mathbb C[x]/(x-c)\to \mathbb C[x]/(x-d)$ a linear map of vector spaces such that $xf(P)=f(xP)$.
Then, $0=(x-d)f(P)=f((x-d)P)=f((x-c+c-d)P)=f((c-d)P)=(c-d)f(P)$, for all $P$.
So, $d=c$ if $f$ is not the $0$ map.
