\section{Properties of Rings}

\begin{theorem}[Rules of Multiplication]
	Let $a,b$, and $c$ belong to a ring $R$. Then
	\begin{enumerate}
		\item $a0 = 0a = 0$.
		\item $a(-b) = (-a)b = -(ab)$.
		\item $(-a)(-b) = ab$.
		\item $a(b-c) = ab - ac$ and $(b-c)a = ba - ca$.
	\end{enumerate}

	Furthermore, if $R$ has a unity element $1$, then

	\begin{enumerate}
		\setcounter{enumi}{4}
		\item $(-1)a = -a$.
		\item $(-1)(-1) = 1$.
	\end{enumerate}
\end{theorem}

\begin{theorem}[Uniqueness of the Unity and Inverses]
	If a ring has a unity, it is unique. If a ring element has a multiplicative inverse, it is unique.
\end{theorem}