I have recently started out learning about fields and the basic field axioms, so I have tried proving a couple of questions related to that. However, when I encountered the question:
Let $F$ be a field, and $a ∈ F$. Then, prove that $a · 0$ = $0$.
I wasn't sure if such an argument (which I mentioned down below) is allowed without further justification.
Basically, to prove this claim, I have used one of the basic axioms which states that "For any $x ∈ F$, there is a $w ∈ F$ such that $x + w = 0$" (introducing the notion of additive inverse). Therefore, I claimed that $$ a⋅0 = (a⋅0) + (-a⋅0) \quad \textrm{(where we have $$x = (a $\cdot$ 0)$ \quad \textrm{and} $ w = -(a $\cdot$ 0))}$$ Thus, $$ a⋅0 = (a⋅0) + (-a⋅0) = 0 \quad \textrm{(by the additive inverse basic axiom)}$$ So, my question is - would this be considered an acceptable argument?