Example 2

Let A be a nonzero ring in which every element is invertible or nilpotent. Then A is a local ring. Indeed, if $a\in A$ is not invertible, then by assumption, there exists an integer $n\geqslant 0$ such that $a^{n+1}= 0$, and $1-a$ has inverse $1 +a+\cdots +a^n$.