\section{Irreducibles, Primes}

\begin{definition}[Associates, Irreducibles, Primes]
	Elements $a$ and $b$ of an integral domain $D$ are called \textit{associates} if $a = ub$, where $u$ is a unit of $D$. A nonzero element $a$ of an integral domain $D$ is called an \textit{irreducible} if $a$ is not a unit and, whenever $b$, $c \in D$ with $a = bc$, then $b$ or $c$ is a unit. A nonzero element $a$ of an integral domain $D$ is called a \textit{prime} if $a$ is not a unit and $a\ \vert\ bc$ implies $a\ \vert\ b$ or $a\ \vert\ c$.
\end{definition}

\begin{theorem}[Prime Implies Irreducible]
	In an integral domain, every prime in an irreducible.
\end{theorem}

\begin{theorem}[PID Implies Irreducible Equals Prime]
	In a principal ideal domain, an element is an irreducible if and only if it is a prime.
\end{theorem}
