Let $R$ and $S$ be two rings. A ring homomorphism between $R$ and $S$ is a function $f:R\to S$ such that:
For all $a$, $b\in R$,
\[f(a+b) = f(a)+f(b).\]For all $a$, $b\in R$,
\[f(ab) = f(a)f(b).\]- \[f(1) = 1.\]
In other words, $f$ is both a group homomorphism from $(R,+)$ to $(S,+)$ and a monoid homomorphism from $(R,\cdot)$ to $(S,\cdot)$.
The kernel of a ring homomorphism $f:R\to S$ is
\[\ker f = f^{-1}[\{0\}].\]
Let $f:R\to S$ is a ring homomorphism.
- \[f(0)=0.\]
For all $a\in R$,
\[f(-a) = -f(a).\]
Proof.By GH#PROP-A.
Let $f:R\to S$ be a ring homomorphism. $f$ is a monomorphism if and only if
\[\ker f = \{0\}.\]
Proof.By GH#PROP-MON.