Created the Abstract Algebra theorems and definitions cheat sheet

2024-01-09 11:30:56 -07:00
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+\chapter{Ideals and Factor Rings}
+\subimport{./}{ideals.tex}
+\subimport{./}{factor-rings.tex}
+\subimport{./}{prime-ideals-and-maximal-ideals.tex}
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+\section{Factor Rings}
+
+\begin{theorem}[Existence of Factor Rings]
+	Let $R$ be a ring and let $A$ be a subring of $R$. The set of cosets $\{r + A\ \vert\ r \in R\}$ is a ring under the operations $(s + A) + (t + A) = s + t + A$ and $(s+A)(t+A)=st+A$ if and only if $A$ is an ideal of $R$.
+\end{theorem}
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+\section{Ideals}
+
+\begin{definition}[Ideal]
+	A subring $A$ of a ring $R$ is called a (two-sided) \textit{ideal} of $R$ if for every $r \in R$ and every $a \in A$ both $ra$ and $ar$ are in $A$.
+\end{definition}
+
+\begin{theorem}[Ideal Test]
+	A nonempty subset $A$ of a ring $R$ is an ideal of $R$ if
+	\begin{enumerate}
+		\item $a-b \in A$ whenever $a,b \in A$.
+		\item $ra$ and $ar$ are in $A$ whenever $a \in A$ and $r \in R$.
+	\end{enumerate}
+\end{theorem}
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+\section{Prime Ideals and Maximal Ideals}
+
+\begin{remark}
+	A \textit{proper} ideal is an ideal $I$ of some ring $R$ such that it is a proper subset of $R$; that is, $I \subset R$.
+\end{remark}
+
+\begin{definition}[Prime Ideal, Maximal Ideal]
+	A \textit{prime ideal} $A$ of a commutative ring $R$ is a proper ideal of $R$ such that $a,b \in R$ and $ab \in A$ imply $a \in A$ or $b \in A$. A \textit{maximal} ideal of a commutative ring $R$ is a \textit{proper} ideal of $R$ such that, whenever $B$ is an ideal of $R$ and $A \subseteq B \subseteq R$, then $B = A$ or $B = R$.
+\end{definition}
+
+\begin{theorem}[$\mathbf{R/A}$ Is an Integral Domain If and Only If $\mathbf{A}$ Is Prime]
+	Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is an integral domain if and only if $A$ is prime.
+\end{theorem}
+
+\begin{theorem}[$\mathbf{R/A}$ Is a Field If and Only If $\mathbf{A}$ Is Maximal]
+	Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is a field if and only if $A$ is maximal.
+\end{theorem}