\section{Definition and Examples}

\begin{definition}[Ring Homomorphism, Ring Isomorphism]
	A \textit{ring homomorphism} $\phi$ from a ring $R$ to a ring $S$ is a mapping from $R$ to $S$ that preserves the two ring operations; that is, for all $a,b$ in $R$,
	\[ \phi(a + b) = \phi(a) + \phi(b)\ \ \ \ \text{and}\ \ \ \ \phi(ab) = \phi(a)\phi(b) \]
	A ring homomorphism that is both one-to-one and onto is called a \textit{ring isomorphism}.
\end{definition}