Does there exist an ideal sheaf $\mathcal F$ on some affine scheme $X$ such that $\mathcal F$ is not quasi-coherent? Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $\mathcal F$ on $X$ such that $\mathcal F$ is not quasi-coherent.

It often helps to translate the problem into commutative algebra.
I will define ideal sheaves on a scheme $X$ to be an $\mathcal{O}_X$ module $\mathcal{I}$ such that for all open sets $U\subset X$, $\mathcal{I}(U)$ is an ideal of $\mathcal{O}_X(U)$. So let $\mathcal{I}$ be an ideal sheaf on $X=\operatorname{Spec}A$. Recall that quasicoherence can be checked by showing that the natural map $\Gamma(\operatorname{Spec}A,\mathcal{I})_f\rightarrow \Gamma(\operatorname{Spec}A_f,\mathcal{I})$ is an isomorphism for all $f$. We clearly have $\Gamma(\operatorname{Spec}A,\mathcal{I})_f=I_f$ where $I$ is some ideal of $A$. Also $\Gamma(\operatorname{Spec}A_f,\mathcal{I})=J$ where $J$ is some ideal of $A_f$. Can you find an ideal $J$ of $A_f$ that doesn't come from localization of an ideal $I$ of $A$?
