Q:

What does it mean for two ideals to coincide locally on a Zariski open subset?

Let $R$ be a Noetherian ring and $I,J$ $R$-ideals. Write
$I^{\left<J\right>}=\bigcup\limits_{n\geq1}(I:J^n)$, where $(I:J^n)=\{r\in R\mid rJ^n\subset I\}$. Show $I^{\left<J\right>}$ is the unique largest $R$-ideal that coincides with $I$ locally on the open set $\operatorname{Spec}(R)\setminus V(J)$
where $\operatorname{Spec}(R)$ is the set of all prime ideals of $R$ with the Zariski topology and $V(J)$ is the set of all prime ideals containing $J$.

My question is what it means for two ideals to coincide locally on a Zariski open subset?

A:

Let $U = \operatorname{Spec} R \setminus V(J)$. I’m guessing that means that for every $\mathfrak p ∈ U$, the localizations of these ideals at $\mathfrak p$ coincide, that is to say $I_{\mathfrak p} = I^{⟨J⟩}_{\mathfrak p}$. At least that’s what is true, if I have checked correctly.

 $I^{⟨J⟩}$ does locally coincide with $I$ in this way:
 
 Obviously, $I ⊆ I^{⟨J⟩}$, so $I_{\mathfrak p} ⊆ I^{⟨J⟩}_{\mathfrak p}$ for all $\mathfrak p ∈ U$.
Let $\mathfrak p ∈ U$. Then $\mathfrak p \not ⊇ J$, so there is some $c ∈ R\setminus \mathfrak p$ with $c ∈ J$. Hence for all $r ∈ I^{⟨J⟩}$ there is some $n ∈ ℕ$ with $rc^n ∈ I$ and so for any $s ∈ R\setminus \mathfrak p$, you have $r/s = rc^n/sc^n ∈ I_\mathfrak p$, so $I^{⟨J⟩}_{\mathfrak p} ⊆ I_{\mathfrak p}$.

I also tried checking that it’s the largest ideal, but I haven’t succeeded. But that wasn’t the question anyway, now was it?

