1. ## Irreducible in PID

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

Proof.

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

Let D be a princple ideal domain and let $p \in D$. Prove that <p> 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 $\left< p \right>$ is a maximal ideal and $p=ab$. Then $\left< p \right> \subseteq \left< a \right>$. Thus either $\left< p \right> \subset \left< a \right>$ or $\left< p \right> = \left< a \right>$. 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 $\left< p \right>$ is maximal) and thus $a$ is a unit.

3. 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 <p> = <a> for any element a in D, then p must contain all the elements in D, thus J = D?