Suppose $a \in \mathbb F, v \in V$, and $av=0$. Prove $a=0$ or $v=0$.

Question

Aside from field and vector space definitions, the only two other propositions I should limit myself to using are $0v =0$ for all $v\in V$ and $a0 =0$ for all $a \in \mathbb{F}$.

I feel like I'm missing something obvious. One implication was straightforward: $$av=a0 \implies a^{-1}av = a^{-1}a0 \implies v =0.$$ As for the other (showing $a=0$ is a possibility) I haven't been able to make much progress. At most I showed that $$(\underbrace{a+a+\cdots +a}_\text{$n$-many})v=0v$$ for positive integers $n \geq 1$, but since $V$ lacks multaplicative inverses it feels like I hit a dead end and can't isolate $a$ somehow.

Could someone at most provide a nudge in the right direction?

Answer 1

Let $av = 0$ where $a \ne 0$ and $v \ne 0$.

Then, in fact we have $a^{-1} av = a^{-1} 0 = 0$, but also $a^{-1} av = (a^{-1}a)v = 1v = v \ne 0$, contradiction.