Do prime ideals expand to prime ideals in the completion?
I believe this is the case since I think $R/P\equiv \hat{R}/P\hat{R}$, although Atiyah-Macdonald explicitly mentions the preservation of quotients only with respect to powers of maximal ideal (I am assuming completion with respect to the maximal ideal here).
EDIT: I guess there is more going on here. If I take $S=k[x]$ with $k$ a field, then $(x+1)S$ is prime in $S$ but $x+1$ is a unit $k[[x]]$. So the modified question is:
When do prime ideals expand to prime ideals in the completion?