Kernel of Ring Epimorphism is Ideal

From ProofWiki
Jump to navigation Jump to search

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


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

$\blacksquare$


Sources