Epimorphism from Division Ring to Ring

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

  • 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 $R = \left\{{0_R}\right\}$.

$\blacksquare$


Sources