# Prime Ideal of Principal Ideal Domain is Maximal

Jump to navigation
Jump to search

## 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.

This article, or a section of it, needs explaining.In particular: Prove the aboveYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

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 $14$