I am being tasked with proving the following statement:
"Let $R$ be an integral domain, let $I,J$ be ideals of $R$, and let $S$ be a multiplicative subset of $R$, with $0 \not \in S$.
Prove that $S^{-1}(I \cap J)=S^{-1}I \cap S^{-1}J$."
I thought that this would be pretty straightforward, but I'm struggling with this.
Could anyone walk me through how a proof for this statement should look?
This is what I've tried:
"Let $\frac{a}{b} \in S^{-1}(I \cap J)$. Then $a \in I \cap J$ and $b \in S$".
Then $a$ can be written in the form $a=i$, where $i \in I$ and $J$."
I'm not sure of how elements in $I \cap J$ should look. I feel like once I have that, then I'll be fine.
For the less trivial part, assume $x\in S^{-1}I\cap S^{-1}J$, so $x=\frac as=\frac bt$ with $a\in I$, $b\in J$, $s,t\in S$. From $\frac as=\frac bt$, we have $x=\frac{at}{st}=\frac{sb}{st}$ where $at=sb\in I\cap J$.