AAR13

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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}}^{+}\,. Deduce that the sum of nilpotent element and a unit is a unit.


If the nilpotent element is 0 then the sum is the unit. If the nilpotent element is a zero divisor for some nonzero element of the ring, multipying the sum by the nonzero element and then by its inverse will give a unit.


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