Let be a commutative ring with unity.
If are distinct maximal ideals of , then
(1) .
(2) .
For (1), the sum of ideals is again an ideal (link). Thus, M+N is an ideal containing M. By hypthesis, M+N should properly contain an maximal ideal M. Thus M+N=A.
For (2), every proper ideal in A is contained in a maximal ideal in A and note that A contains the unity (link).
Assume is a proper ideal in A. Then, should be contained in a maximal ideal. It follows that should be contained in either M or N (check their intersection). Contradiction !
Thus, is an ideal in A which is not a proper ideal in A. We conclude that .