\part{Rings and Subrings}


\chapter{Rings}

Groups are powerful algebraic structures, but they only consider one operation. We don't yet have the language to talk about how two operations interact with eachother. A few notions that we're familiar with from elementary algebra and elementary number theory are not adequately generalized by group theory alone.

Ring theory develops the necessary theory to model algebraic structures with two operations, however it still draws heavily on group theory.

\section{Rings}

\begin{definition}
A \emph{ring} is a 3-tuple $(R,+, \cdot )$ of a set $G$ and a binary operation $\cdot$ acting on $G$ with the following properties:
\begin{itemize}
	\item $(R,+)$ is an Abelian group 
	\item $(R,\cdot)$ is a monoid
	\item $\cdot$ is distributive with respect to $+$
\end{itemize}

\begin{itemize}
	\item $R$ is the set of elements the ring works over
	\item $+ : R \times R \to R$ is the 'addition' operator of the ring
	\item $\cdot : R \times R \to R$ is the 'multiplication' operator of the ring
\end{itemize}
We write $0_R$ as the identity element of $(R,+)$ and $1_R$ the identity element of $(R,\cdot)$
\end{definition}

When the operations are apparent, a ring $(R,+,\cdot)$ may be denoted as $R$, and the multiplication expression $r\cdot s$ may be contracted to $rs$. Note that when speaking generally about rings, 'addition' refers to the ring operation forming the Abelian group and 'multiplication' refers to the ring operator forming the monoid.

We will denote inverse elements with repect to addition by a minus sign rather than our familiar '-1 exponent' (it'll be clear soon why), so the additive inverse of $r$ is denoted as $-r$.


\section{Properties of rings}
It is useful to understand the fundamental properties of rings before proceeding further with ring theory.

\begin{proposition}Let $(G,\cdot)$ be a group:
\begin{itemize}
	\item $0_R$ is an absorbing element; multiplication with $0_R$ is always $0_R$
\end{itemize}
\[ \forall r \in R \]
\[ r 0_R = 0_R r = 0_R\]
\[(-1_R)r= r(-1_R) = -r\]
\[-(rs)=r(-s)=(-r)s\]
\end{proposition}
\begin{proposition}

\end{proposition}

\begin{proposition}
If $1_R=0_R$, then $R$ is the trivial ring; the only element in the ring is $0_R$.
\end{proposition}

Many concepts from group theory translate easily to set theory.

\begin{definition}
The \emph{order} of a ring is the cardinality of its set.
\end{definition}



\begin{definition}
A \emph{finite ring} is a ring with finite order.
\end{definition}


\begin{definition}
A \emph{commutative ring} is a ring such that multiplication is commutative.
\end{definition}




\begin{definition}
A \emph{subring} of $R$ is a ring $S$ such that $(S,+)$ is a subgroup of $(R,+)$ and $(S,\cdot)$ is a subgroup of $(R,\cdot)$.
\end{definition}




That said, there are some unique concepts in ring theory that describe the relationship between its Abelian group and monoid.


\begin{definition}
An element $x \in R \setminus \{0\}$ of a ring is a \emph{zero divisor} iff there is a $y \in R \setminus \{0\}$ such that $xy=yx=0$.
\end{definition}


Multiplication in rings is a monoid rather than a group, so there is no guarantee of inverse elements. However, it is possible that some elements are invertible with respect to multiplication; such elements are called \emph{units}.

\begin{definition}
An element $x \in R$ of a ring is a \emph{unit} iff there is a $y\in R$ $xy=yx=1_R$. For a group $R$, denote its set of units as $R^{*}$.
\end{definition}


Like groups, inverse elements are unique (when they exist). We will denote the multiplicative inverse of the unit $r$ as $r^{-1}$.

\begin{proposition}
If $r$ is a unit of $R$, there exists a unique $r^{-1}$ such that $rr^{-1}=r^{-1}r=1_R$
\end{proposition}


\section{Special classes of rings}

\begin{definition}
A \emph{domain} $D$ is a ring with no zero divisors.
\[ D \text{ is a domain } \iff D \text{ is a ring } \land \{ d \in D\setminus \{0_D\}  : \exists e [ de=ed=0_D] \}=\emptyset\]
\end{definition}

\begin{proposition}[Cancellation law for domains]
Let $(D,+,\cdot)$ be a domain, if $a\neq 0_R$ and $ax=ay$ then $x=y$.
\end{proposition}


\begin{definition}
A \emph{field} $F$ is a ring with where all elements except for $0_F$ are units.
\[ F \text{ is a field } \iff F \text{ is a ring } \land F^{*}=F \setminus \{0_F \}\]
\end{definition}



\begin{proposition}
A ring $F$ is a field iff $(F,\cdot)$ is an Abelian group.
\end{proposition}


\begin{proposition}
Fields are domains.
\end{definition}

