Let $R$ be a commutative ring and $S$ a subring of $R$. If $I$ is an ideal of $S$ define $I^e$ as the ideal in $R$ generated by $I$, i.e. the extension of $I$ in $R$.
If $I,J$ are ideals in $S$, we can form the ideal quotient $(I:J)=\{x\in S:xJ\subseteq I\}$ in $S$ and then extend this ideal, resulting in $(I:J)^e$. Or we can first extend $I$ and $J$ and then form the ideal quotient in $R$, resulting in $(I^e:J^e)$.
Now, I can show that the former ideal is a subset of the latter, that is, $(I:J)^e\subseteq (I^e:J^e)$, but I have not been able to prove the reverse inclusion, nor have I found an example showing that the latter ideal can be strictly larger. Any help is appreciated.
Take $S=k[x,y],\quad R=k[x,y,z]/(yz-x)$. Take $I=(x),J=(y)$.
Note that $z\in(I^e:J^e)\setminus(I:J)^e$.
Or an alternative formulation of the same counterexample: Take $k[x,y]\subset k[y,\frac{x}{y}]$, with the same ideals - $I=(x), J=(y)$, and observe $x/y$.