I have managed to prove that the following quotient groups are isomorphic as abelian groups, but I am struggling to prove that they are not isomorphic as $\Bbb Q[x]$-modules.
$M = \Bbb Q[x]/⟨x^2 + 1⟩$
$N= \Bbb Q[x]/⟨x^2 -1⟩.$
My main struggle is what should I consider as an operation for them to be modules? Or should I just prove it for the natural operation (multiplication of the $\Bbb Q[x]$ polynomial ring?) and if so what happens when the operation is abstract?
On
Given any commutative ring $R$ and any two ideals $I$ and $J$ of $R$, if $R/I$ and $R/J$ are isomorphic as $R$-modules, then one must have $I=\mathrm{Ann}(R/I)=\mathrm{Ann}(R/J)=J$.
So, $\mathbb{Q}[x]/(x^2+1)$ and $\mathbb{Q}[x]/(x^2-1)$ are not isomorphic as $\mathbb{Q}[x]$-modules, as they have different annihilators.
I mean... multiplying any element in $M$ by $x^2 + 1$ yields zero, whereas doing the same for $1 \in N$ yields $2 \neq 0$. Unless I am missing something, this proves they are not isomorphic as $Q[x]$ modules.