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Let $R$ be a Noetherian local ring, $\mathrm{gr}(R)$ be its associated graded ring. I know that when $R$ is regular, then $\mathrm{gr}(R)$ is a domain (in fact a polynomial ring). I also know an example where $R$ is a domain but $\mathrm{gr}(R)$ is not: say $R=\mathbb C[x,y]/(y^2-x^3-x^2)$, the coordinate ring of a nodal curve. However in this example $R$ is not normal (i.e. integrally closed).

I am wondering if $R$ is normal implies $\mathrm{gr}(R)$ is a domain.

user26857
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Yifeng Huang
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    $R=\mathbb C[X,Y,Z]_{(X,Y,Z)}/(X^2-Y^2Z-Z^3)$ with $\mathrm{gr}(R)=\mathbb C[X,Y,Z]/(X^2)$. Maybe a little too complicated, but anyway... we need $\dim R\ge2$. – user26857 Apr 02 '17 at 23:10

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