# Quotient Ring of Kernel of Ring Epimorphism

## Theorem

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

Let $K = \map \ker \phi$ be the kernel of $\phi$.

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

$g \circ q_K = \phi$

## Proof

From the Quotient Theorem for Epimorphisms, there is one and only one isomorphism that satisfies the conditions for each of the operations on $R_1$.

Hence the result.

$\blacksquare$