Irreducible in PID

**tttcomrader** · November 25th 2007, 09:41 AM

Let D be a princple ideal domain and let $p \in D$ . Prove that is a maximal ideal in D iff p is irreducible.

Proof.

Assume p is irreducible, and let p = ab, then either

**ThePerfectHacker** · November 25th 2007, 10:03 AM

Originally Posted by tttcomrader

Let D be a princple ideal domain and let $p \in D$ . Prove that is a maximal ideal in D iff p is irreducible.

(Actually we should also specify that $p$ is not a unit also).
Here is half of the argument:
Say that $\left$ is a maximal ideal and $p=ab$ . Then $\left \subseteq \left< a \right>$ . Thus either $\left \subset \left< a \right>$ or $\left = \left< a \right>$ . If the first case then $p$ and $a$ are associate elements so $b$ is a unit. If the second case then $\left< a \right> = D$ (because $\left$ is maximal) and thus $a$ is a unit.

**tttcomrader** · November 25th 2007, 12:22 PM

Conversely, assume p = ab, with b a unit.

Suppose $ \subseteq J \subseteq D$ , I need to show that J = D.

Can I say that since = <a> for any element a in D, then p must contain all the elements in D, thus J = D?

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