Let be a nilpotent element of the commutative ring . Let for minimal . 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.