Epimorphism from Division Ring to Ring

Theorem

Let $\struct {K, +, \circ}$ be a division ring whose zero is $0_K$.

Let $\struct {R, +, \circ}$ 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, $\map \ker K$ must therefore either be $0_K$ or $K$.

Let $\map \ker K = 0_K$.

Then by Ring Epimorphism with Trivial Kernel is Isomorphism $\phi$ is an ring isomorphism.

Thus $R$ a division ring like $K$.

Let $\map \ker K = K$.

Then:

$\forall x \in K: \map \phi x = 0_R$

As $\phi$ is an epimorphism, it is surjective.

Therefore:

$R = \set {0_R}$

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.3$