Ideals of Division Ring

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

The only ideals of $\left({R, +, \circ}\right)$ are $\left\{{0_R}\right\}$ and $R$ itself.

Proof

 * From Null Ideal, $\left\{{0_R}\right\}$ is an ideal of $\left({R, +, \circ}\right)$, as $\left({R, +, \circ}\right)$, being a division ring, is also a ring.

By definition, every non-zero element of a division ring is a unit.

So $S \ne \left\{{0_R}\right\} \implies \exists x \in S: x \ne 0_R$ such that $r$ is a unit of $R$.

The result follows from Ideal of Unit is Whole Ring.