# Definition:Kernel of Ring Homomorphism

## Definition

Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings.

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:

$\map \ker \phi = \set {x \in R_1: \map \phi x = 0_{R_2} }$

where $0_{R_2}$ is the zero of $R_2$.

That is, $\map \ker \phi$ is the subset of $R_1$ that maps to the zero of $R_2$.

From Ring Homomorphism Preserves Zero it follows that $0_{R_1} \in \map \ker \phi$ where $0_{R_1}$ is the zero of $R_1$.