In this post I’ll describe the structure theorem over PID’s which generalizes the following results:

• Finite dimensional vector fields over $k$ are all of the form $k^{\oplus n}$,
• The classification theorem for finitely generated abelian groups,
• The Frobenius normal form of a matrix,
• The Jordan decomposition of a matrix.

1. Some ring theory prerequisites

(Prototypical example for this section: $R = \mathbb Z$.)

Before I can state the main theorem, I need to define a few terms for UFD’s, which behave much like $\mathbb Z$: Our intuition from the case $R = \mathbb Z$ basically carries over verbatim. We don’t even need to deal with prime ideals and can factor elements instead.

Definition 1. If $R$ is a UFD, then $p \in R$ is a …

The following is an excerpt from a current work of mine. I thought I’d share it here, as some people have told me they enjoyed it.

As I’ll stress repeatedly, a matrix represents a linear map between two vector spaces. Writing it in the form of an $m \times n$ matrix is merely a very convenient way to see the map concretely. But it obfuscates the fact that this map is, well, a map, not an array of numbers.

If you took high school precalculus, you’ll see everything done in terms of matrices. To any typical high school student, a matrix is an array of numbers. No one is sure what exactly these numbers represent, but they’re told how to magically multiply these arrays to get more arrays. They’re told that the matrix

$$(\begin{array}{cccc}1& 0& \dots & 0\\ 0& 1& \dots & 0\\ \mathrm{⋮}& \mathrm{⋮}& \ddots & \mathrm{⋮}\\ 0& \mathrm{0\; \dots }\end{array}$$