app_ring.tex

\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