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Let $x$ be a nonzero element in a commutative ring, then $\exists y\neq0(xy=0)$ ($x$ is a zero divisor) iff $\exists y\neq 0(x^2y=0)$. $(\rightarrow)$ part is pretty trivial. How to prove the other way?
Hint: $x^2y=x(xy)$. Consider the cases $xy = 0$ and $xy \ne 0$.
Required, but never shown