A ring homomorphism between two rings $R$ 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 GRP-HOM > Proposition 1.
Let $f : R \to S$ be a ring homomorphism. $f$ is a monomorphism if and only if
\[\ker f = \{0\}.\]
Proof. By GRP-HOM > Proposition 5.