# Jordan form of a matrix and finitely generated modules over PIDs.

We know that any square matrix of order n over Complex numbers is similar to a Jordan form,I am told that this relates to the structure theorem for finitely generated modules over PIDs. Can anyone give me some hint of how these two things relate to each other?

• The Jordan canonical form is more related to the primary decomposition of a module over a PID, than the one that usually occurs in the structure theorem. – Joppy Mar 7 at 10:16
• Can you please explain a little more?,i mean,why do we need exactly PIDs? – Soumyadip Sarkar Mar 7 at 11:40
• The polynomial ring $k[x]$ (where $k$ is a field) is a PID. A module over this ring is precisely a $k$-vector space $V$ equipped with a linear map $T: V \to V$, i.e. the action of $x$. If $V$ is finite-dimensional, then $(V, T)$ is finitely generated as a $k[x]$-module, and so the structure theorem for modules of PIDs applies. But this is not a good place to learn about them - you should consult an algebra textbook. – Joppy Mar 7 at 11:42
• Sure,i will look it up,,thanks for your concern. – Soumyadip Sarkar Mar 7 at 13:34