\section{Subrings}

\begin{definition}[Subring]
	A subset $S$ of a ring $R$ is a \textit{subring of $R$} if $S$ is itself a ring with the operations of $R$.
\end{definition}

\begin{theorem}[Subring Test]
	A nonempty subset $S$ of a ring $R$ is a subring if $S$ is closed under subtraction and multiplication -- that is, if $a - b$ and $ab$ are in $S$ whenever $a$ and $b$ are in $S$.
\end{theorem}
