Examples of decomposition of torsion modules

Question

I am reading Lang’s Algebra. On page 149 there is a theorem:

Let $E$ be a finitely generated torsion module (not 0). Then E is the direct sum: $$ E=\bigoplus_{p} E(p) $$ Where $p$ is prime and $E(p) \neq 0$ . Each $E(p)$ can be written as a direct sum: $$ E(p)=\bigoplus_i R/({p}^{v_i}) $$

I’ve known that the theorem applies to prove the Smith normal form for matrices. But are there any other examples besides matrices and the case when modules reduce to abelian groups? In fact it is hard for me to imagine what the theorem is demonstrating in general.

I would appreciate it if anyone gives more (nontrivial) examples, hopefully with explanations on understanding this theorem.

Answer 1

This theorem is the torsion part of the general structure theory for modules over $PID$s, and it’s true that the most important examples of these are $\mathbb{Z}$ and $k[x]$. The first of these gives abelian groups, and the second gives linear algebra, via interpreting $x$ as an endomorphism of a vector space.

Here are some applications in linear algebra, that are all corollaries of the structure theorem for torsion modules over $k[x]$. We say that an endomorphism of a vector space is semisimple if the associated $k[x]$ module is a direct sum of simple modules.

1. Any matrix $M$ can be expressed uniquely as $M_{ss}+M_n$, where $M_{ss}$ is semisimple, $M_n$ is nilpotent, and these commute. Furthermore, these matrices can both be expressed as polynomials in $M$.

2. For any monic polynomial $p(x)$ of degree $n$, there is a semisimple matrix with $p$ as it’s characteristic polynomial, and this semisimple representative is unique up to conjugacy.

3. If two matrices with coefficients in a field $k$ are conjugate via a matrix with coefficients in a field extension $K$, then they are conjugate via a matrix with entries in $k$.

4. The structure of the centraliser of any element in $GL_n(k)$ is very clear through the torsion module perspective.

5. Any matrix is similar to its transpose.

These results are all nontrivial, and many are used extensively in the theory of algebraic groups. They can all be proven without too much difficulty, once one has the structure theorem for torsion modules over $k[x]$, which is essentially a classification of matrices up to conjugation. These are often proven by the Jordan canonical form, but even this is just a direct corollary of the classification of torsion modules!