If $ R $ is a commutative ring with unity and $ a,b \in R $, then how do I prove that $$ a \neq 0, ~ b \neq 0 ~~ \Longrightarrow ~~ a \cdot b \neq 0? $$
Does this also hold for any ring?
It’s not true in general: consider $a=2,b=3$ in the commutative ring $\Bbb Z/6\Bbb Z$, which is unital. Rings of the kind that you want are integral domains.