Ring Homomorphism

Definition 1. A ring homomorphism between two rings $R$ and $S$ is a function $f : R \to S$ such that:

1. For all $a$, $b\in R$,

   \[f(a + b) = f(a) + f(b).\]

2. For all $a$, $b \in R$,

   \[f(ab) = f(a)f(b).\]

In other words, $f$ is both a group homomorphism from $(R, +)$ to $(S, +)$ and a monoid homomorphism from $(R, \cdot)$ to $(S, \cdot)$.

Definition 2. A ring homomorphism $f : R \to S$ is neutral-preserving if

\[f(1_R) = 1_S.\]

Definition 3. The kernel of a ring homomorphism $f : R \to S$ is

\[\ker f = f^{-1}[\{0\}].\]

Proposition 1. For any ring homomorphism $f : R \to S$:

1. \[f(0) = 0.\]

2. For all $a \in R$,

   \[f(-a) = -f(a).\]

Proposition 2. Let $f : R \to S$ be a ring homomorphism. $f$ is a monomorphism if and only if

\[\ker f = \{0\}.\]