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