\section{Ideals}

\begin{definition}[Ideal]
	A subring $A$ of a ring $R$ is called a (two-sided) \textit{ideal} of $R$ if for every $r \in R$ and every $a \in A$ both $ra$ and $ar$ are in $A$.
\end{definition}

\begin{theorem}[Ideal Test]
	A nonempty subset $A$ of a ring $R$ is an ideal of $R$ if
	\begin{enumerate}
		\item $a-b \in A$ whenever $a,b \in A$.
		\item $ra$ and $ar$ are in $A$ whenever $a \in A$ and $r \in R$.
	\end{enumerate}
\end{theorem}