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Answer by Slade for Examples of rings whose polynomial rings have large...
Here is a classification of all integral domains $A$ with $\dim A=1$, $\dim A[X]=3$:First, consider the map $\operatorname{Spec} A[X]\to\operatorname{Spec} A$ given by $\mathfrak{p}\to\mathfrak{p}\cap...
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If $A$ is a commutative ring with unity, then a fact proved in most commutative algebra textbooks is:$$\dim A + 1\leq\dim A[X] \leq 2\dim A + 1$$Idea of proof: each prime of $A$ in a chain can arise...
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