\definition{Ring} A ring $\langle R, +, \cdot \rangle$ is a set $R$ together with two binary operations $+$ and $\cdot$, which we call additoin and multiplication, defined on $R$ such that te following axioms are satisfied:
\begin{itemize}
\item $\langle R, +\rangle$ is an abelian group.
\item Multiplication is associative.
\item For all $a, b, c \in R$, the {\bf left distributive law,} $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ and the {\bf right distributive law} $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$ hold.
\end{itemize}