Epimorphism from Division Ring to Ring

Theorem
Let $\left({K, +, \circ}\right)$ be a division ring whose zero is $0_K$.

Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0_R$.

Let $\phi: K \to R$ be a ring epimorphism.

Then one of the following applies:


 * $(1): \quad R$ is a null ring
 * $(2): \quad R$ is a division ring and $\phi$ is a ring isomorphism.

Proof
We have that the kernel of $K$ is an ideal.

From Ideals of a Division Ring, $\ker \left({K}\right)$ must therefore either be $0_K$ or $K$.


 * If $\ker \left({K}\right) = 0_K$, then by Ring Epimorphism with Trivial Kernel is Isomorphism $\phi$ is an ring isomorphism, thus making $R$ a division ring like $K$.


 * If $\ker \left({K}\right) = K$, then $\forall x \in K: \phi \left({K}\right) = 0_R$.

As $\phi$ is an epimorphism, it is surjective and therefore $R = \left\{{0_R}\right\}$.