Quotient Ring of Kernel of Ring Epimorphism

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.

Let $K = \ker \left({\phi}\right)$.


Then 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

From the Quotient Theorem for Epimorphisms, there is one and only one isomorphism that satisfies the conditions for each of the operations on $R_1$.

So the first statement follows directly.


From Kernel is Trivial iff Monomorphism, $\phi$ is a ring monomorphism iff $K = \left\{{0_{R_1}}\right\}$.

As $\phi$ is also an epimorphism, the result follows.


Sources