# Maximal Ideals Question

• March 24th 2012, 11:10 PM
sashikanth
Maximal Ideals Question
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?