$1-r$ unit in ring with $r^n = 0$

Question

Let $R$ be a ring with $r \in R$ and $r^n = 0$ for $n \in \mathbb{N}$. Show that $1-r$ is a unit in $R$.

I tried to use the geometric sum but I dont know how to proceed.

That technique works. Note that it is a finite sum by your assumption. — Randall, Nov 19 '18 at 15:27

score 3 · Accepted Answer · answered Nov 19 '18 at 15:29

3

We have $(1-r)(1+r+\ldots+r^{n-1}) = 1$ if $r^n=0$.

answered Nov 19 '18 at 15:29

Wuestenfux

1

This is one of the most duplicated questions on the site. Please next time duplicate it as such instead of answering Thanks. – rschwieb Nov 19 '18 at 15:32

1 Answers1