I'm reading the notes on Elliptic Curves from this MIT course, more specifically this part where the local ring of a curve $C$ at point $P$ is defined, as the set of rational functions $f$ on $C$ such that $f(P) \neq \infty$.
It is stated (see §23.3) that a local ring is a principal ideal domain, without a proof.
I can see that it is a domain, but I don't see why it should be principal (e.g. every ideal from this local ring is generated by a single element).
As this is something important for my understanding of those notes, I would like to have a proof of that fact. I would especially appreciate an answer that does not involve too many abstractions besides what is included in those notes.
An answer showing how to exhibit a generator of an ideal of a local ring on a particular example would also be appreciated.