Let $A \subset B$ a ring extension. It is well known that if the extension is integral, then $\dim B=\dim A$. I can think of some examples where the Krull dimension increases (and by that I mean $\dim B>\dim A$), like $A \subset A[X]$ for any ring $A$. And this situation looks a bit natural or intuitive to me, even though Krull dimension has nothing to do with others notions of dimension I'm used to work with, like the dimension of a vector space. In the few exemples of extension decreasing the Krull dimension I've seen, $B$ is a field.
So here is my question:
can you give some examples of extensions decreasing Krull dimension where $B$ is not a field?