Created the Abstract Algebra theorems and definitions cheat sheet

2024-01-09 11:30:56 -07:00
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+\chapter{Introduction to Rings}
+\subimport{./}{motivation-and-definition.tex}
+\subimport{./}{properties-of-rings.tex}
+\subimport{./}{subrings.tex}
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+\section{Motivation and Definition}
+
+\begin{definition}[Ring]
+	A \textit{ring} $R$ is a set with two binary operations, addition (denoted by $a + b$) and multiplication (denoted by $ab$), such that for all $a,b,c$ in $R$:
+	\begin{enumerate}
+		\item $a + b = b + a$.
+		\item $(a + b) + c = a + (b + c)$.
+		\item There is an additive identity 0. That is, there is an element 0 in $R$ such that $a + 0 = a$ for all $a$ in $R$.
+		\item There is an element $-a$ in $R$ such that $a + (-a) = 0$.
+		\item $a(bc) = (ab)c$.
+		\item $a(b+c) = ab + ac$ and $(b + c)a = ba + ca$.
+	\end{enumerate}
+\end{definition}
+
+\begin{remark}
+	Note that multiplication need not be commutative. When it is, we say that the ring is \textit{commutative}. Also, a ring need not have an identity under multiplication. A \textit{unity} (or \textit{identity}) in a ring is a nonzero element that is an identity under multiplication. A nonzero element of a com-
+	mutative ring with unity need not have a multiplicative inverse. When it does, we say that it is a unit of the ring. Thus, $a$ is a unit if $a^{-1}$ exists.
+
+	\noindent The following terminology and notation are convenient. If $a$ and $b$ belong to a commutative ring $R$ and $a$ is nonzero, we say that $a$ \textit{divides} $b$ (or that $a$ is a \textit{factor} of $b$) and write $a \vert b$, if there exists an element $c$ in $R$ such that $b = ac$. If $a$ does not divide $b$, we write $a \nmid b$.
+\end{remark}
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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}
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+\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}