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

Proof

Existence of Kernel

By Kernel of Ring Homomorphism is Ideal:

The kernel of $\phi$ is an ideal of $R_1$.

$\Box$

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

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 22$: Theorem $22.6: \ 1^\circ$