Let be a commutative ring and let be nilpotent – that is, for some positive integer . Prove the following.
- is either zero or a zero divisor.
- is nilpotent for all .
- is a unit in .
- The sum of a unit and a nilpotent element is a unit.
- Say is minimal such that . If , then . If , then , and , so that is a zero divisor.
- Since is commutative, we have .
- Note that . Thus is a unit.
- Let be a unit and nilpotent. Then is nilpotent, so is a unit, and thus is a unit.