I apologise for my incompleteness. This whole exercise aims to prove that $F[X, X^{-1}]$ is a PID, so can most certainly not use that fact.. In fact, the question arose from reading this related post: math.stackexchange.com/questions/590219/…

But this results from a far more general result: the localisation of a P.I.D. is itself a P.I.D. (that's why I wrote ‘hence’ in my answer).

Yes, absolutely.

@Bernard Localisation is beyond the scope of my (first introduction to ring theory) course.

@reuns Thanks, I totally forgot $X^{-1}$ was a unit here. By $\mathfrak{a}$, do you mean any element or subset of $F[X,X^{-1}]$?. By $\bar{J}$ do you mean $\tilde{J}$ or the set generated by $J$?

The Fraktur font is the usual font for ideals, especially when there's a risk to confuse with elements of a ring. By $J$, I mean like you: the ideal generated by $\tilde J$.

You speak of $\bar{J}$ though. Also, do you then mean that $(f, \mathfrak{a})$ is "the ideal generated by the element $f$ and the ideal $\mathfrak{a}$"? I'm not sure what that means... Is it "the ideal generated by the set $\mathfrak{a} \cup \{f\}$"?

On second thoughts, this doesn't seem to get me that much closer. Are you sure this tells me that $j$ is expressible as a single multiple $X^{-\deg^-(f)}f$ for some $f \in I$?

Previous question: $(f,\mathfrak a)$denotes the sum of the ideal generated by $f$ and the ideal $\mathfrak a$. Last question: don't you agree that $j$ and $f_0$ generate the same ideal in $F[X,X^{-1}]$? That's the main point of my suggestion

@JosvanNieuwman: Are you supposed to know that $K[X,X^{-1}]$ is a noetherian ring?

I'm here!

Sorry, no :(. I appreciate your help but it's also night here now, so do you mind if I come back to you in about 10 hours..?

Maybe in the morning it will all become clear (what you've said)

It's late here too. I'm not sure I'll have much time in 10 hours, but you may try. I'll probably have more time in, say 18-20 hours (I'll be back home).

Great, thanks a lot!!

You're welcome. Good night!

Good day

I've been looking at it again with one of my teachers, and I want to ask you the following

Consider the last part of the proof here:math.stackexchange.com/questions/590219/…

The part where he proves that $f \in (j) \Rightarrow f \in I$. In our opinion, what he does is unnecessary. Since j generates $(J)$, we already know it must be of the form

$r_{f_1}X^{-deg^-(f_1)}f_1 + ... + r_{f_n}X^{-deg^-(f_n)}f_n$, where all $f_i \in I$ and $r_{f_i} \in F[X]$. This clearly is in I, so $f = gj \in I$