Let $R$ be a finite ring that satisfies the following conditions:
(1) For any $x\in R$, if $x\ne 0$ then $x^2\ne 0$.
(2) There exists at most one nonzero element $y\in R$ that satisfies $y^2=y$.
How can we show that $R$ has no zero divisors?
Let $R$ be a finite ring that satisfies the following conditions:
(1) For any $x\in R$, if $x\ne 0$ then $x^2\ne 0$.
(2) There exists at most one nonzero element $y\in R$ that satisfies $y^2=y$.
How can we show that $R$ has no zero divisors?