## Constructing units from nilpotent elements in a commutative ring

Let $R$ be a commutative ring with $1 \neq 0$. Prove that if $a \in R$ is nilpotent, then $1-ab$ is a unit for all $b \in R$.

By this previous exercise, $\mathfrak{N}(R)$ is an ideal of $R$. Thus for all $b \in R$, $-ab$ is nilpotent. By this previous exercise, $1 - ab$ is a unit in $R$.