chap2.tex

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\section{Rings}
\begin{example}[Examples of a ring]
	Fields, Reals, Complexes, Integers, Gaussian Integers, Polynomial Rings and Matrix Rings.
\end{example}
\begin{definition}
	A \textit{Ring} $R$ satisfies the following properties
	\begin{enumerate}
		\item $R$ forms an abelian group under addition
		\item $R$ forms an group under multiplication.
		\item The distributive property holds, i.e.,
		\[ a(b + c) = ab + ac\] 
	\end{enumerate}
\end{definition}
\begin{example}
	Analogy between groups and Rings (Fields)
	\begin{enumerate}
		\item Set $S$ acted on by group $G$ $\iff$ vector space $V$ over field $\b F$ acted on by rings of matrices.
		\item Symmetric group $\iff$ matrix rings = all linear transformation of vector space.
		\item Union of sets $\iff$ Direct sum $V \oplus W$ of vector spaces.
		\item Product of sets $A \times B$ $\iff$ Tensor product of spaces $V \otimes W$
 	\end{enumerate}
\end{example}
\begin{example}[More Exotic Rings]
	\url{https://en.wikipedia.org/wiki/Burnside_ring}
\end{example}
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