Field has no Proper Zero Divisors/Proof 1

Theorem

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Then $\struct {F, +, \times}$ has no proper zero divisors.

That is:

$a \times b = 0_F \implies a = 0_F \lor b = 0_F$

Proof

By definition, $F$ is a division ring.

Again by definition, a division ring is a ring with unity with no proper zero divisors.

$\blacksquare$