# Kernel of Ring Epimorphism is Ideal

## Theorem

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

Then:

### Kernel of Ring Homomorphism is Ideal

The kernel of $\phi$ is an ideal of $R_1$.

### Quotient Ring of Kernel of Ring Epimorphism

There exists a unique ring isomorphism $g: R_1 / K \to R_2$ such that:

$g \circ q_K = \phi$

### Ring Epimorphism with Trivial Kernel is Isomorphism

$\phi$ is an isomorphism if and only if $K = \set {0_{R_1} }$.