Maximal Ideal of Division Ring

Theorem
Let $\left({D, +, \circ}\right)$ be a Division Ring whose zero is $0$.


 * Let $\left({J, +, \circ}\right)$ be a maximal ideal of $D$.

Then $J = \left\{{0}\right\}$.

Proof
From Ideals of Division Ring, the only ideals of a Division Ring $\left({D, +, \circ}\right)$ are $\left({D, +, \circ}\right)$ and $\left({\left\{{0}\right\}, +, \circ}\right)$.

Hence the result by definition of maximal ideal.