
Maximal Ideals Question
Q) Let $\displaystyle I, J $ be two distinct maximal ideals of a commutative ring $\displaystyle R $ with $\displaystyle 1 $. Show that $\displaystyle I+J=R $
My solution:
Since the sum of two ideals is an ideal, we know that $\displaystyle I+J $ is an ideal of $\displaystyle R $
Now $\displaystyle I \subseteq I+J \subseteq R $
Since $\displaystyle I $ is maximal, we have either $\displaystyle I+J=I $ or $\displaystyle I+J=R $
But since it is given that $\displaystyle I, J $ are distinct, we cannot have$\displaystyle I+J=I $, hence the result follows.
My question: I have not used the fact that $\displaystyle R $ has $\displaystyle 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?