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.