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app_ring.tex
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app_ring.tex
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\section{Abstract Ring}\label{app:ring}
\begin{definition}
A ring $\Ring$ is a set with two binary operators: addition $+\colon
\Ring\times\Ring\to\Ring$ and multiplication $\cdot\colon\Ring\times\Ring\to\Ring$ such that
\begin{enumerate}
\item $\Ring$ is an Abelian group under addition,
\begin{itemize}
\item (closure) for any $a,b \in \Ring$: $a+b\in\Ring$;
\item (associativity) for any $a,b,c \in \Ring$: $a+(b+c) = (a+b)+c$;
\item (neutral) there exists $0\in \Ring$ such that for any $a\in\Ring$:
$a+0 = 0+a = a$;
\item (inverse) for any $a\in\Ring$ there exists $-a \in\Ring$ such
that $a+(-a)=(-a)+a = 0$;
\item (commutativity) for any $a,b\in\Ring$: $a+b=b+a$.
\end{itemize}
\item $\Ring$ is a monoid under multiplication,
\begin{itemize}
\item (closure) for any $a,b \in \Ring$: $a\cdot b\in\Ring$;
\item (associativity) for any $a,b,c \in \Ring$: $a\cdot(b\cdot c) =
(a\cdot b)\cdot c$;
\item (neutral) there exists $1\in \Ring$ such that for any $a\in\Ring$:
$a\cdot 1 = 1\cdot a = a$.
\end{itemize}
\item addition and multiplication are compatible,
\begin{itemize}
\item (left distributivity) for any $a,b,c\in\Ring$: $a\cdot(b+c) = a\cdot
b+a\cdot c$;
\item (right distributivity) for any $a,b,c\in\Ring$: $(b+c)\cdot a =
b\cdot a + c\cdot a$.
\end{itemize}
\end{enumerate}
\end{definition}
% note: If the multiplication was an Abelian group, then the ring becomes a field