Let $k$ be a field, not necessarily algebraically closed. Then how would you show that the extension $k[x] \subset k[x,y]$ does or does not satisfy Going-Up?
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No, that extension does not satisfy Going-Up.
Indeed the associated morphism $f:\mathbb A^2_k \to \mathbb A^1_k$ is the first projection and is well known not to be a closed map:
What has this got to do with Going-Up? A lot! Atiyah-Macdonald prove in (Exercise 6.11, page 79) that a morphism of rings $A\to B$ with $B$ noetherian satisfies Going-Up if and only if the associated morphism of affine schemes $Spec(B)\to Spec(A)$ is closed.
The direct way Alternatively, we can atttack the problem frontally.