Ideals of Division Ring

Theorem

Let $\struct {R, +, \circ}$ be a division ring whose zero is $0_R$.

The only ideals of $\struct {R, +, \circ}$ are $\set {0_R}$ and $R$ itself.

That is, $\struct {R, +, \circ}$ has no non-null proper ideals.

Proof

From Null Ring is Ideal, $\set {0_R}$ is an ideal of $\struct {R, +, \circ}$, as $\struct {R, +, \circ}$, being a division ring, is also a ring.

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

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

The result follows from Ideal of Unit is Whole Ring.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.2$: Corollary