Please help demystify the classification of finite modules over PIDs for me. I want to decompose a finitely generated module over a PID into its elementary divisors or its invariant factors (see Dummit & Foote, Theorems 12.5 and 12.6 if you're not familiar).
Here's an example from Artin:
Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2 = 0$.
How does one start this process in the above simple case? I understand much of the proofs in the text, but can't seem to work out a simple example like this one.
I see the same textbook problem here: Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2=0$
But don't believe they've completed the step of decomposing into its elementary divisors or invariant factors. Can someone help with this particular step?
A module over the ring $R=\mathbb{C}[x]/\langle x^2\rangle$ is a complex vector space $V$ with a linear operator $T$ such that $T^2=0$.
If $V$ is finitely generated over $R$, then $V$ is finite dimensional over $\mathbb{C}$. (Why?)
So, the question really is:
This question is now in a familiar form for which you can use the classification of finitely generated modules over PIDs.
The invariant factor decomposition will be $T \mid T \cdots \mid T^2 \cdots \mid T^2$. The number of factors of each type will determine the module. This is of course the same as a decomposition of the matrix of $T$ into a block for $\ker T$ plus some $2 \times 2$ Jordan blocks.