Classifying finitely generated modules over ring

How to characterize every finitely generated module over $$\mathbb{Q}[X]/(X^2+1)^3$$?

I feel somehow we should use the structure theorem for modules over PID but the ring here is not a domain. So I am lost for ideas.

• We just can say they're quotients of a finite number of copies of the ring (as a module over itself). May 7 '20 at 22:26
• If $R$ is a (commutative) ring, and $I$ is an ideal of $R$, then an $R/I$-module is nothing more than an $I$-torsion $R$-module, so it suffices to classify finitely generated $\mathbb{Q}[X]$-modules which are $(X^{2}+1)^{3}$-torsion. May 7 '20 at 22:27

Let $$M$$ be a finitely generated $$\mathbb Q[x]/ (x^2+1)^3$$- module. By definition, we have a ring hom $$\mathbb Q[x]/ (x^2+1)^3 \to \mathrm{End}_{\mathrm{Ab}}(M)$$.
This is the same as a ring hom $$\mathbb Q[x] \to \mathrm{End}_{\mathrm{Ab}}(M)$$ where $$(x^2+1)^3$$ is contained in the kernel of the action. Therefore, a $$\mathbb Q[x]/ (x^2+1)^3$$-module, $$M$$, is the same as a $$\mathbb Q[x]$$-module where $$(x^2+1)^3 \cdot M = 0$$.
$$\mathbb Q[x]$$ is a PID and since $$M$$ is a finitely generated over $$\mathbb Q[x]/ (x^2+1)^3$$, then it is finitely generated over $$\mathbb Q[x]$$.