# Tagged: idempotent element

## Problem 543

Let $R$ be a ring with $1$.
Suppose that $a, b$ are elements in $R$ such that
$ab=1 \text{ and } ba\neq 1.$

(a) Prove that $1-ba$ is idempotent.

(b) Prove that $b^n(1-ba)$ is nilpotent for each positive integer $n$.

(c) Prove that the ring $R$ has infinitely many nilpotent elements.