Prime Ideal of Principal Ideal Domain is Maximal

Theorem

Let $D$ be a principal ideal domain whose zero is $0_D$.

Let $J \subseteq D$ be a nonzero prime ideal.

Then $J$ is maximal.

Proof

As $D$ is a principal ideal domain, every ideal of $D$ is a principal ideal $\ideal r$ generated by some $r \in D$.

So, let $\ideal p$ be an arbitrary prime ideal of $D$ generated by $p$, where $p \ne 0_R$.

As $\ideal p$ is prime, $p$ is irreducible.

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The result follows from Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $14$