Let $R$ be a finite commutative ring with no zero-divisors. We have proven that then $R$ has a unity.
Is it then always true that $R = (y)$ when $y \neq 0$?
We have a function $ x \mapsto xy$ where $y \neq 0$ from $R \to (y)$. Then the kernel must be zero, thus making it injective, and a bijection since both are finite. Does this always hold?