# Minimal prime divisors (MinAss R)

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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What does the equality "Min Ass R = Ass R" exactly mean ? – Ralph Feb 23 '12 at 3:42
It would be nice to have a definition of MinAss here... – darij grinberg Feb 23 '12 at 3:43
MinAss means minimal primes in Ass(R). "Min Ass R = Ass R" means R has no embedded prime ideals. – Mahdi Majidi-Zolbanin Feb 23 '12 at 4:04
MinAss means minimal primes in Ass(R). "Min Ass R = Ass R" means R has no embedded prime ideals – Stella Feb 23 '12 at 8:49

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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oh dear, I don't understated language of this paper. Can you explain it more. – Stella Feb 24 '12 at 13:14
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/… – Mahdi Majidi-Zolbanin Feb 24 '12 at 14:23
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. – Mahdi Majidi-Zolbanin Feb 24 '12 at 16:01