Exercise 1.1

Question

Let $x$ be a nilpotent element of a ring $A$.
  Show that $1 + x$ is a unit of $A$.
  Deduce that the sum of a nilpotent element and a unit is a unit.

Amit Awasthi · Accepted Answer

\begin{proof}
  Suppose $x^n = 0$, and let $y = \sum_{i=0}^{n-1} (-x)^i$.
  Then, $(1+x)y = 1$.
  Now if $u \in A^	imes$, we see $(u^{-1}x)^n = 0$.
  Thus, $u+x = u(1+u^{-1}x)$ is a unit since each factor is.
\end{proof}

Exercise 1.1

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Exercise 1.1

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