Lately I've been fascinated by the result that one might state slightly informally
Lemma. (In the context of linear algebra over a field.) If $p$ and $q$ are relatively prime polynomials and $T$ is a linear operator then $\ker(pq(T))=\ker(p(T))\oplus\ker(q(T))$.
Follows easily from the fact that $F[x]$ is a PID; you can use it to start a proof of the existence of the Jordan Canonical Form, also for a proof that the solution to a constant-coefficient linear homogeneous DE is what it is.
Q: Does this result have a standard name? Or do we know who proved it?
In French schools and universities it is called "Lemme des Noyaux" which is a translation of Kernel Lemma. It is one of the rare occasions where I find a page inthe french wikipedia with no english counterpart.
Here is a link to said page : https://fr.wikipedia.org/wiki/Lemme_des_noyaux
I suspect it has its own page because it is a standard result in the linear algebra curriculum. Sadly i haven't found a name in english but calling it the Kernel Lemma is a good idea.