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Hello All,is This conclusion true?

If $(R,m)$ is a local ring and $ Min Ass R=Ass R$ then can we conclude that $Min Ass \hat{R}=Ass \hat{R}$? ($\hat{R}$ is $m$-adic completion of $R$)

$MinAss$ means minimal primes in $Ass(R)$. "$Min Ass R = Ass R$" means that $R$ has no embedded prime ideals. In fact, if every associated prime ideal of $R$ is minimal then every associated prime ideal of $\hat{R}$ is minimal?

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  • $\begingroup$ What does the equality "Min Ass R = Ass R" exactly mean ? $\endgroup$
    – Ralph
    Feb 23, 2012 at 3:42
  • $\begingroup$ It would be nice to have a definition of MinAss here... $\endgroup$
    – darij grinberg
    Feb 23, 2012 at 3:43
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    $\begingroup$ MinAss means minimal primes in Ass(R). "Min Ass R = Ass R" means R has no embedded prime ideals. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Feb 23, 2012 at 4:04
  • $\begingroup$ MinAss means minimal primes in Ass(R). "Min Ass R = Ass R" means R has no embedded prime ideals $\endgroup$
    – Stella
    Feb 23, 2012 at 8:49

1 Answer 1

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The answer is no in general. In the paper Fibres formelles d'un anneau local noethérien D. Ferrand and M. Raynaud give an example of a two-dimensional local domain whose $\mathfrak{m}$-adic completion has embedded prime ideals. In the same paper, they mention that the answer is yes in certain special cases, such as, when $R$ is a quotient of a Cohen-Macaulay ring, or when $R$ is universally Japanese.

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  • $\begingroup$ oh dear, I don't understated language of this paper. Can you explain it more. $\endgroup$
    – Stella
    Feb 24, 2012 at 13:14
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    $\begingroup$ If by language you mean French, then you can look at the paper Local domains with bad sets of formal prime divisors By Brodmann and Rotthaus: sciencedirect.com.rpa.laguardia.edu:2048/science/article/pii/… $\endgroup$
    – Mahdi Majidi-Zolbanin
    Feb 24, 2012 at 14:23
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    $\begingroup$ The link I posted above doesn't seem to work. Brodmann and Rotthaus paper is published in Journal of Algebra, Volume 75, Issue 2, April 1982, Pages 386–394. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Feb 24, 2012 at 16:01

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