\begin{definition}[Definition:Ring Homomorphism]
Let $\struct {R, +, \circ}$ and $\struct{S, \oplus, *}$ be rings.
Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$.
That is, $\forall a, b \in R$:
{{begin-eqn}}
{{eqn | n = 1
      | l = \map \phi {a + b}
      | r = \map \phi a \oplus \map \phi b
}}
{{eqn | n = 2
      | l = \map \phi {a \circ b}
      | r = \map \phi a * \map \phi b
}}
{{end-eqn}}
Then $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is a ring homomorphism.
\end{definition}