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

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


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

