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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?

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    @GeorgesElencwajg, don't worry about that. I understood as such.2012-03-15

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No, this is not true: prime ideals do not extend to prime ideals after completion.

Consider $S=\mathbb C[X,Y]$ and its localization $R=S_M$at $M=(X,Y)\subset S$, so that $R=\mathbb C[X,Y]_{(X,Y)}$ is a local ring with maximal ideal $\mathfrak m=(X,Y)R$.
The completion of $R$ along $\mathfrak m$ is the ring of formal power series $\hat R=\mathbb C[[X,Y]]$.
The principal prime ideal $\mathfrak p=(Y^2-X^2-X^3)\subset R$ has as extension the principal ideal $\hat {\mathfrak p}=\mathfrak p\hat R=(Y^2-X^2-X^3)\hat R$ which is no longer prime:
Indeed $Y^2-X^2-X^3=(Y+X\sqrt {1+X})(Y-X\sqrt {1+X})\in \hat {\mathfrak p}\;$ although $Y+ X\sqrt {1+X},Y-X\sqrt {1+X}\notin \hat{\mathfrak p}$