Created the Abstract Algebra theorems and definitions cheat sheet

part-3/chapters/chapter-13/chapter-13.tex

@@ -0,0 +1,4 @@
+\chapter{Integral Domains}
+\subimport{./}{definition-and-examples.tex}
+\subimport{./}{fields.tex}
+\subimport{./}{characteristic-of-a-ring.tex}

part-3/chapters/chapter-13/characteristic-of-a-ring.tex

@@ -0,0 +1,13 @@
+\section{Characteristic of a Ring}
+
+\begin{definition}[Characteristic of a Ring]
+	The \textit{characteristic} of a ring $R$ is the least positive integer $n$ such that $nx = 0$ for all $x$ in $R$. If no such integer exists, we say that $R$ has characteristic 0. The characteristic of $R$ is denoted by $\characteristic R$.
+\end{definition}
+
+\begin{theorem}[Characteristic of a Ring with Unity]
+	Let $R$ be a ring with unity 1. If 1 has infinite order under addition, then the characteristic of $R$ is 0. If 1 has order $n$ under addition, then the characteristic of $R$ is $n$.
+\end{theorem}
+
+\begin{theorem}[Characteristic of an Integral Domain]
+	The characteristic of an integral domain is 0 or prime.
+\end{theorem}

part-3/chapters/chapter-13/definition-and-examples.tex

@@ -0,0 +1,13 @@
+\section{Definition and Examples}
+
+\begin{definition}[Zero Divisors]
+	A \textit{zero-divisor} is a nonzero element $a$ of a commutative ring $R$ such that there is a nonzero element $b \in R$ with $ab = 0$.
+\end{definition}
+
+\begin{definition}[Integral Domain]
+	An \textit{integral domain} is a commutative ring with unity and no zero-divisors.
+\end{definition}
+
+\begin{theorem}[Cancellation]
+	Let $a,b$, and $c$ belong to an integral domain If $a \neq 0$ and $ab = ac$, then $b = c$.
+\end{theorem}

part-3/chapters/chapter-13/fields.tex

@@ -0,0 +1,13 @@
+\section{Fields}
+
+\begin{definition}[Field]
+	A \textit{field} is a commutative ring with unity in which every nonzero element is a unit.
+\end{definition}
+
+\begin{theorem}[Finite Integral Domains are Fields]
+	A finite integral domain is a field.
+\end{theorem}
+
+\begin{corollary}[$\mathbf{\Z_p}$ Is a Field]
+	For every prime $p$, $\Z_p$, the ring of integers modulo $p$ is a field.
+\end{corollary}