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 $K = \map \ker \phi$.

Then there is a unique ring isomorphism $g: R_1 / K \to R_2$ such that:


 * $g \circ q_K = \phi$

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

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

So the first statement follows directly.

From Kernel is Trivial iff Monomorphism, $\phi$ is a ring monomorphism $K = \set {0_{R_1} }$.

As $\phi$ is also an epimorphism, the result follows.