It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with respect to the ring homomorphism taking any $r\in R$ to $r/1$).
I have three questions:
1) Is any nilpotent (nil) ideal of $R_S$ the extension of a nilpotent (resp. nil) ideal of $R$?
2) Is any idempotent ideal of $R_S$ the extension of an idempotent ideal of $R$?
3) May it be possible that extension of two distinct ideals of $R$ in $R_S$ (with respect to the homomorphism above) be equal?
It is not difficult to see that any nilpotent (idempotent) ideal extends to a nilpotent (idempotent) ideal. I think the answer to the second question is yes when $R$ is a domain.
Any point of suggestion is appreciated!