Ideals in Laurent polynomials over a field

Question

Be $F$ a field and let $I$ be any ideal in $F[X,X^{-1}]$. For any $f = \sum_{n\in \mathbb{Z}}a_nX^n \in F[X,X^{-1}]$, define deg$^-(f) := \min\{n \in \mathbb{Z} \mid a_n \neq 0\}$. Consider the set $\tilde{J} := \{X^{-\deg^-(f)}f \mid f \in I\}$ and let $J$ be the ideal generated by $\tilde{J}$ in $F[X]$. As $F[X]$ is a PID, we can let $j \in F[X]$ be such that $J = (j) = jF[X]$. I want to show that $jF[X,X^{-1}] \subseteq I$. For any $f \in jF[X,X^{-1}]$ there is a $g \in F[X,X^{-1}]$ such that $f = gj$. Now if $j$ were to be of the form $X^{-\deg^-(f')}f'$ for some $f' \in I$, we would be done. However, I can only see $j$ as a linear combination of elements of $I$ (with coefficients in $F[X]$), but that doesn't make it a multiple of any one such element.

Answer 1

$F[X,X^{-1}]$ is the localisation $F[X]_X$, and hence is a P.I.D. Let $f_0$ a generator of the ideal $I$.

Claim: a generator of $J$ in $F[X]$ is $$j=X^{-\deg^{-}(f_0)}f_0. $$

Indeed, any Laurent polynomial $f$ in $I$ can be written as $\;f=f_0\,g,\enspace g\in F[X,X^{-1}]$. Note that $\;\deg^{-}(f)=\deg^{-}(f_0)+\deg^{-}(g) $, so that $$X^{-\deg^{-}(f)}f=X^{-\deg^{-}(f_0)}f_0\,X^{-\deg^{-}(g)}g(X^{-\deg^{-}(g)}g)j.$$