TITLE: Prime and Primary Ideals in Completion of a ring
QUESTION [5 upvotes]: Let $(R,\mathfrak m)$ be a local noetherian ring and $\widehat{R}$  its $\mathfrak m$-adic completion. 

If $\mathfrak q\in \operatorname{Spec}(\widehat{R})$ then can we find $\mathfrak p\in  \operatorname{Spec}(R)$ such that $\mathfrak q=\mathfrak p\widehat{R}$ ?
Is a primary ideal of  $\widehat{R}$   an extension of a primary ideal of  $R$ ?

REPLY [3 votes]: This is false: take $R$ a local noetherian integral domain such that $\hat R$ is not an integral domain and $\mathfrak q$ a minimal ideal of $\hat R$. Since $R\to\hat R$ is (faithfully) flat $\mathfrak q$ is lying over $(0)$ and $\mathfrak q\neq(0)$. 
Obviously, from 1., this is also false. However one can say something positive: there is a one-to-one correspondence between the set of $\mathfrak m$-primary ideals in $R$ and the set of $\hat{\mathfrak m}$-primary ideals in $\hat R$ given by $\mathfrak a\to\mathfrak a\hat R$.