Q) Let  I, J be two distinct maximal ideals of a commutative ring  R with  1 . Show that  I+J=R

My solution:

Since the sum of two ideals is an ideal, we know that  I+J is an ideal of  R

Now  I \subseteq I+J \subseteq R

Since  I is maximal, we have either  I+J=I or  I+J=R

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

My question: I have not used the fact that  R has  1 . 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?