Kernel of Ring Epimorphism is Ideal

Theorem

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring epimorphism.

Then:

Kernel of Ring Homomorphism is Ideal

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

Quotient Ring of Kernel of Ring Epimorphism

There exists a unique ring isomorphism $g: R_1 / K \to R_2$ such that:

$g \circ q_K = \phi$

Ring Epimorphism with Trivial Kernel is Isomorphism

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.6: \ 1^\circ$