Which of the following inequalities hold for a ring $R$ and an ideal $I\subset R$?
$\operatorname{height}I\leq\dim R-\dim R/I$
$\operatorname{height}I\geq\dim R-\dim R/I$
Which of the following inequalities hold for a ring $R$ and an ideal $I\subset R$?
$\operatorname{height}I\leq\dim R-\dim R/I$
$\operatorname{height}I\geq\dim R-\dim R/I$
Assume $\dim R$ is finite. Let $P$ be a prime containing $I$ such that $\dim R/I = \dim R/P$. We then have
$$\operatorname{ht}I + \dim R/I = \operatorname{ht}I + \dim R/P\le \operatorname{ht}P + \dim R/P \le \dim R.$$
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