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

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 exists a unique ring isomorphism $g: R_1 / K \to R_2$ such that $g \circ q_K = \phi$
 * $\phi$ is a ring isomorphism iff $K = \left\{{0_{R_1}}\right\}$.