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Which of the following inequalities hold for a ring $R$ and an ideal $I\subset R$?

$\operatorname{height}I\leq\dim R-\dim R/I$

$\operatorname{height}I\geq\dim R-\dim R/I$

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Assume $\dim R$ is finite. Let $P$ be a prime containing $I$ such that $\dim R/I = \dim R/P$. We then have

$$\operatorname{ht}I + \dim R/I = \operatorname{ht}I + \dim R/P\le \operatorname{ht}P + \dim R/P \le \dim R.$$

Reference

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    Right, you're right2011-07-11
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I think to have it: suppose $\mathrm{height}\;I=n$ and $\mathrm{dim}\;R/I=m$ then we have a chain

$\mathfrak{p}_0\subset\ldots\subset\mathfrak{p}_n\subset I\subset\mathfrak{p}_{n+1}\subset\ldots\subset\mathfrak{p}_{n+m}$

but in general $\mathrm{dim}\;R$ would be greater, so

$\mathrm{height}\;I+\mathrm{dim}\;R/I\leq\mathrm{dim}\;R$ holds