I'm trying to understand this example:

$M= \mathbb{C}[x]/(x^2)$ is a module over $\mathbb{C}[x]$, (since if R is any ring and I is any ideal of R, then R/I is always an R-module.)

Then

(i) $M$ is cyclic

(ii) $M$ is not a free module over $\mathbb{C}[x].$

I understand the first claim. $M$ is cyclic because the element $1 \in \mathbb{C}[x]$ generates $M:$ $\mathbb{C}[x]/(x^2)= \mathbb{C}[x]\cdot 1$.

In class, the explanation given for why $M$ is not a free module is that $x^2\cdot1= 0,$ and $x^2 \neq 0,$ so the module can't be free.

I understand that for a module to be free, it must have a basis. So if $X$ were a basis, and $1\in X,$ then the above statement would make sense. But why is $1$ necessarily in the basis? What am i missing?