Let $R$ and $S$ be two rings. A ring homomorphism between $R$ and $S$ is a function $f:R\to S$ such that:
- \[f(a+b) = f(a)+f(b)\]
for all $a$, $b\in R$.
- \[f(a\cdot b) = f(a)\cdot f(b)\]
for all $a$, $b\in R$.
- \[f(1) = 1.\]
Namely, $f$ is both a group homomorphism from $(R,+)$ to $(S,+)$ and a monoid homomorphism from $(R,\cdot)$ to $(S,\cdot)$.
DEF-RH-TRI. Trivial Ring Homomorphism.
The trivial / zero ring homomorphism between two rings $R$ and $S$ is the function
\[f : R\to S, \, a\mapsto 0.\]
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.
- \[f(0)=0.\]
- \[f(-a) = -f(a)\]
for every $a\in R$.
Proof.By PROP-GH-A.
Let $f:R\to S$ be a ring homomorphism. $f$ is a monomorphism if and only if
\[\ker f = \{0\}.\]
Proof.By PROP-GH-MON.