Kernel of Ring Epimorphism is Ideal

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

Then:
 * The kernel of $$\phi$$ is an ideal of $$R_1$$.


 * 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
The proof of these assertions can be found on the following pages:

Existence of Kernel
By Kernel of Ring Homomorphism is Ideal:
 * The kernel of $$\phi$$ is an ideal of $$R_1$$.

Uniqueness of Quotient Mapping
By Quotient Ring of Kernel of Ring Epimorphism:
 * 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 $$K = \left\{{0_{R_1}}\right\}$$.