The fundamental unit in the ring of algebraic integers. 1

Question

Let $R$ be a ring. Suppose that there exists an element $r\in R$ with $r^n = 0$, for some $n \geq 1$. Prove that $1 - r$ is invertible. May I know how we can prove this theory with some examples?

Answer 1

\begin{align*} (1-r)(1 + r + r^2 + ... + r^{n-1}) &= 1 + r + r^2 + ... + r^{n-1} -r -r^2 - ... - r^{n-1} - r^n \\ &= 1 \end{align*}