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.

$\blacksquare$

Sources

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