Classifying finitely generated modules over ring

Question

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.

Answer 1

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 $\mathbb Q[x]$.

Now you can use the classification theorem for finitely generated modules over PIDs.