Ring theory/Ring homomorphism/Definition

Ring homomorphism

Let ${}R$ and ${}S$ be rings. A mapping

$\varphi \colon R\longrightarrow S$

is called a ring homomorphism, if the following properties hold:

1. ${}\varphi (a+b)=\varphi (a)+\varphi (b)$.
2. ${}\varphi (1)=1$.
3. ${}\varphi (a\cdot b)=\varphi (a)\cdot \varphi (b)$.