Quotient Ring of Kernel of Ring Epimorphism

Theorem
Let $$\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$$ be a ring epimorphism.

Let $$K = \ker \left({\phi}\right)$$.

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 iff $$K = \left\{{0_{R_1}}\right\}$$.

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 of Monomorphism is Trivial, $$\phi$$ is a ring monomorphism iff $$K = \left\{{0_{R_1}}\right\}$$.

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