Division Ring has No Proper Zero Divisors

Theorem

Let $\struct {R, +, \circ}$ be a division ring.

Then $\struct {R, +, \circ}$ has no proper zero divisors.

Proof

Let $\struct {R, +, \circ}$ be a division ring whose zero is $0_R$ and whose unity is $1_R$.

By definition of division ring, every element $x$ of $R^* = R \setminus \set {0_R}$ has an element $y$ such that:

$y \circ x = x \circ y = 1_R$

That is, by definition, every element of $R^*$ is a unit of $R$.

The result follows from Unit of Ring is not Zero Divisor.

$\blacksquare$

Sources

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