Let be R a commutative ring whit unit and let I a proper ideal of R. Let R' the completion of R respect to the ideal I (see Introduction to Commutative Algebra - M. F. Atiyah, I. G. MacDonald for the definition). It's true or false that:



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> R' is an integral domain then R is an integral domain?
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The completion of a ring R is a domain then the ring R is a domain?
No. $\mathbb{Z}/6$ completed at the prime $(3)$ is $\mathbb{Z}/3$. (I can think of true versions of this statment, but I'll hold off until it's clearer what you're looking for.)


