A commutative ring $R$ is local if and only if, for all $r,s\in R$, $r+s=1$ implies $r$ or $s$ is a unit.

Question

A commutative ring $R$ is local if and only if, for all $r,s\in R$, $r+s=1$ implies $r$ or $s$ is a unit.

How to start upon this problem?

Answer 1

Hint: If $r$ is not a unit, what do you know about $(r)$? And if neither is a unit, then what does $r+s=1$ tell you?