One problem:

A commutative ring R with identity is local iff for all r,s in R, r+s=1_{R}implies r or s is a unit.

I've proved the easy part of the proof (=>), but for the converse one? So far, I only have a result that if K is a set consisting of all nonunits of R, then for any u, v in K, u+v won't be 1_{R}. Does it help? Any suggestion?