Basic properties of nilpotent ring elements

Let R be a commutative ring and let x \in R be nilpotent – that is, x^n = 0 for some positive integer n. Prove the following.

  1. x is either zero or a zero divisor.
  2. rx is nilpotent for all r \in R.
  3. 1+x is a unit in R.
  4. The sum of a unit and a nilpotent element is a unit.

  1. Say m is minimal such that x^m = 0. If m = 1, then x = 0. If m > 1, then x \neq 0, x^{m-1} \neq 0 and x \cdot x^{m-1} = 0, so that x is a zero divisor.
  2. Since R is commutative, we have (rx)^m = r^mx^m = 0.
  3. Note that (1 - (-x))(\sum_{i=0}^{m-1} (-x)^i) = (\sum_{i=0}^{m-1} (-x)^i) - (\sum_{i=0}^{m-1} (-x)^{i+1}) = (\sum_{i=0}^{m-1} (-x)^i) - (\sum_{i=1}^{m} (-x)^i = 1 + (\sum_{i=1}^{m-1} (-x)^i) - (\sum_{i=1}^{m-1} (-x)^i) - (-x)^m = 1 - (-1)^mx^m = 1. Thus 1+x is a unit.
  4. Let u be a unit and x nilpotent. Then u^{-1}x is nilpotent, so 1+u^{-1}x is a unit, and thus u(1+u^{-1}x) = u+x is a unit.
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