Constructing units from nilpotent elements in a commutative ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.27
Solution:
By Exercise 7.3.29, $\mathfrak{N}(R)$ is an ideal of $R$. Thus for all $b \in R$, $-ab$ is nilpotent. By Exercise 7.1.14, $1 - ab$ is a unit in $R$.