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Let x\, be a nilpotent element of the commutative ring R\,. Let x^{m}=0\, for minimal m\in {\mathbb  {Z}}^{+}\,. Prove that x\, is either zero or a zero divisor.

x^{m}=x(x^{{m-1}})=0\, so x=0\, or x\, is a zero divisor of the nozero element x^{{m-1}}\,.

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