In a given field $F$, if $ab=0$, is it true that at least $a$ or $b$ has to be zero?

Question

The answer is true for integral domains, but I wonder if it is also true for any general field. Thanks.

Answer 1

Without loss of generality, assume ${a\neq 0}$. Then ${a}$ has a multiplicative inverse. Multiply both left hand sides by this multiplicative inverse:

$${\Rightarrow a^{-1}\left(ab\right)=0}$$ $${\Rightarrow b=0}$$

So indeed, at least one of the elements must be $0$.

Answer 2

Yes. Every field is an integral domain.