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) $$R$$ is a null ring;
 * 2) $$R$$ is a division ring and $$\phi$$ is a ring isomorphism.

Proof
The kernel of $K$ is an ideal.

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


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


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

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