Local rings characterization According to Wikipedia, a (non-commutative) ring $R$ is local if and only if there do not exist two proper (principal) (left) ideals $I_1, I_2$ such that $R = I_1 + I_2$. It is easy to see that local rings have this property, but I'm having trouble showing the other way: if there do not exist two proper co-maximal (principal) (left) ideals then the ring is local.
HINT $\ $ If $\rm\ R \ne 0\ $ is nonlocal then there exists an $\rm\: r\:$ such that both $\rm\: r\:$ and $\rm\: 1-r\:$ are nonunits. Choose maximal ideals containing them.
HINT $\ $ If $\rm\ R \ne 0\ $ is nonlocal then there exists an $\rm\: r\:$ such that both $\rm\: r\:$ and $\rm\: 1-r\:$ are nonunits. Choose maximal ideals containing them.