Dimension of a PID

Mar 17, 2025 3:07 PM

Let $A$ be a commutative ring with $1$ . The dimension of $A$ is the maximum length $d$ of a chain of prime ideals $p_{0} ⊊ p_{1} ⊊ \dots ⊊ p_{d}$ . Prove that if $A$ is a PID, the dimension of $A$ is at most 1.

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Let $A$ be a commutative ring with $1$. The {\bfseries dimension} of $A$ is the maximum length $d$ of a chain of prime ideals $\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \cdots \subsetneq \mathfrak{p}_d$. Prove that if $A$ is a PID, the dimension of $A$ is at most 1.