\section{Prime Ideals and Maximal Ideals}

\begin{remark}
	A \textit{proper} ideal is an ideal $I$ of some ring $R$ such that it is a proper subset of $R$; that is, $I \subset R$.
\end{remark}

\begin{definition}[Prime Ideal, Maximal Ideal]
	A \textit{prime ideal} $A$ of a commutative ring $R$ is a proper ideal of $R$ such that $a,b \in R$ and $ab \in A$ imply $a \in A$ or $b \in A$. A \textit{maximal} ideal of a commutative ring $R$ is a \textit{proper} ideal of $R$ such that, whenever $B$ is an ideal of $R$ and $A \subseteq B \subseteq R$, then $B = A$ or $B = R$.
\end{definition}

\begin{theorem}[$\mathbf{R/A}$ Is an Integral Domain If and Only If $\mathbf{A}$ Is Prime]
	Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is an integral domain if and only if $A$ is prime.
\end{theorem}

\begin{theorem}[$\mathbf{R/A}$ Is a Field If and Only If $\mathbf{A}$ Is Maximal]
	Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is a field if and only if $A$ is maximal.
\end{theorem}
