Integral extensions preserve Krull dimension
= Integral extensions preserve Krull dimension
If $A\subseteq B$ is an <integral extension>, then $\dim A=\dim B$. The <incomparability theorem for integral extensions> contracts every strict prime chain in $B$ to a strict chain in $A$, giving $\dim B\leq\dim A$. Conversely, the <Lying-over theorem> starts over the bottom of any prime chain in $A$, and repeated use of the <going-up theorem> lifts the whole chain to $B$.