Q) Let be two distinct maximal ideals of a commutative ring with . Show that

My solution:

Since the sum of two ideals is an ideal, we know that is an ideal of

Now

Since is maximal, we have either or

But since it is given that are distinct, we cannot have , hence the result follows.

My question: I have not used the fact that has . Is my proof incorrect/incomplete? As far as I know, maximal ideals can be defined for commutative rings without unity too, so that data has not been given for the purpose of rigour. What is it's use?