Field/Domain/Fact/Proof

We prove this by contradiction, so we assume that ${\displaystyle {}a}$ and ${\displaystyle {}b}$ are both not ${\displaystyle {}0}$. Then there exist inverse elements ${\displaystyle {}a^{-1}}$ and ${\displaystyle {}b^{-1}}$, and hence ${\displaystyle {}(ab){\left(b^{-1}a^{-1}\right)}=1}$. On the other hand, we have ${\displaystyle {}ab=0}$ by the premise, and so the annihilation rule gives

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

hence ${\displaystyle {}0=1}$, which contradicts the field properties.