Let $\hat{R}$ is $m$-adic completion of a local ring $(R,m)$.What is the relation between $Min R$ and $Min \hat{R}$. we know that $\hat{R}$ is faithfully flat $R$-module.
$Min R$=set of all minimal prime divisors of zero.
I think 


$p\in Min(R)$ iff $\hat{p}=p\hat{R}\in Min\hat{R}$.



relation between Min(R)  and Min(R^)
What you are sayinng is not true in general. For example, the completion of a domain may not be a domain. An example is given in exercises of Bourbaki's Commutative Algebra.


