Maximal Ideal of Division Ring

Theorem
Let $\struct {D, +, \circ}$ be a Division Ring whose zero is $0$.

Let $\struct {J, +, \circ}$ be a maximal ideal of $D$.

Then:
 * $J = \set 0$

Proof
From Ideals of Division Ring, the only ideals of a Division Ring $\struct {D, +, \circ}$ are $\struct {D, +, \circ}$ and $\struct {\set 0, +, \circ}$.

Hence the result by definition of maximal ideal.