## Boolean Rings Do Not Have Nonzero Nilpotent Elements

## Problem 618

Let $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a **Boolean ring**.)

**(a)** Prove that $x^n=x$ for any positive integer $n$.

**(b)** Prove that $R$ does not have a nonzero nilpotent element.