Let $(A,+,•)$ be a ring with unity, which is not a division ring, and $|A|=25.$ Let $a$ be the number of divisors of $0$ in this ring. Then a) $a=1;$ b) $a=2;$ c) $a=3;$ d) $a \geq 4.$
I guess that the answer is d), since this would be the case if we let $A=\mathbb{Z}/25 \mathbb{Z}.$ However, I couldn't prove it for general $A.$
I know that a finite ring with $0 \neq 1$ which doesn't have divisors of $0$ is a division ring. Hence, our $A$ must have divisors of 0. I tried to use the characteristic and I got that it can only be $5$ or $25,$ but this is where I got stuck.
If $e\neq 0$ is a left zero divisor in $A$, then the set of zero divisors contains the left ideal $Ae$. Since this ideal is non-trivial (it contains $e$) and it is not equal to $A$ (it doesn't contain $1$) it is of size $5$. Thus you have at least $4$ zero divisors.