Extension of ideals in integral extensions

Question

Let $R\subset S$ be an integral extension in the category of commutative rings with unity. I have three questions:

1) Is every ideal of $S$ an extended ideal?
2) Is extension of each idempotent ideal of $R$ an idempotent ideal of $S$?
3) Is extension of each (nil) nilpotent ideal of $R$ a (nil) nilpotent ideal of $S$?

Since here we have the inclusion as the ring homomorphism from $R$ to $S$, for each ideal $I$ in $R$ any element of $I^e$ is of the form of a finite sum $\sum s_ir_i$ with $s_i\in S, r_i\in R$. So, I think the answer to the third question is in affirmative.

Answer 1

1. No. Consider the extension $\mathbb Z\subset\mathbb Z[i]$. Then the ideal $(1+i)$ is not the extension of any ideal. (If $n\mathbb Z[i]=(1+i)\mathbb Z[i]$ then $n$ and $1+i$ are associates, so $n^2=N(n)=N(1+i)=2$.)

2. $(IS)^2=I^2S=IS$.

3. $(IS)^n=I^nS=0$ provided $I^n=0$.