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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:

R' is an integral domain then R is an integral domain?

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    $\begingroup$ Well, you need some hypothesis since completion only sees things locally. (Take the disjoint union of two lines $R = k[x] \times k[y]$ and complete wrt a maximal ideal on one of the lines but not the other). $\endgroup$ – LMN Jan 8 '13 at 19:01
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    $\begingroup$ Is $R$ a local ring? $\endgroup$ – Karl Schwede Jan 8 '13 at 20:52
  • $\begingroup$ hm - I remember being given this as a homework problem, and given the way it's stated it certainly smells like one... $\endgroup$ – Kevin Casto Jan 9 '13 at 6:41
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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.)

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  • $\begingroup$ Please, could you elaborate on "I can think of true versions of this statement"? $\endgroup$ – user237522 Feb 16 '20 at 8:23
  • $\begingroup$ @user237522 If $R$ is a noetherian local ring, then $R$ injects in $R'$. So, in this situation, if $R'$ is an integral domain, then so is $R$. $\endgroup$ – David E Speyer Feb 16 '20 at 13:11
  • $\begingroup$ Thank you very much! $\endgroup$ – user237522 Feb 16 '20 at 20:06

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