Created the Abstract Algebra theorems and definitions cheat sheet

part-3/chapters/chapter-12/properties-of-rings.tex

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+\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}