I recently learned about free $R$-modules. I saw some non-examples, including $\mathbb{Z}/2\mathbb{Z}$ in $\mathbb{Z}$ and $\mathbb{Q}[t]/(t^2-1)$ in $\mathbb{Q}[t]$.
Is it true that, in general, $R/I$ is not free for any nontrivial ideal $I$ of $R$?
This seems to be the case, because if $I$ is not trivial, there is some nonzero $x\in I$ such that the equivalence class of $x$ in $R/I$ cannot be written as a unique sum of elements in $R$. (I realise a more precise/careful argument is probably needed). Is this idea correct, or have I misunderstood something?
I'll assume $R$ is commutative.
The annihilator of a (nonzero) free module is always zero.
The annihilator of $R/I$ is $I$.