# Math Help - Ring Proof Problem

1. ## Ring Proof Problem

Here's the question:

An element $\displaystyle{a}$ in a non-trivial ring $R$ is called nilpotent if $a^k = 0$ for some integer $k \geq 1$. Prove that if a is nilpotent then a-1 has a multiplicative inverse. Give an example to illustrate this in $Z_{18}$.

I've absolutely no idea how to start this question. I'd be very grateful for any help

2. You can handle this problem using something of a dirty trick. From the real numbers, we have the following formula:

$\frac{1}{1-a}=1+a+a^2+\ldots+a^{k-1}+a^k+\ldots$

This suggests that we try to show that $(1-a)^1=1+a+a^2+\ldots+a^k$. Try carrying it out to see what you get.

Good luck.

3. Should that be $(1-a)^{-1}$?