Suppose $I_1$ is a maximal ideal in $C^*$ algebra $A$,$I_2$ is an ideal of $A$,then $I_1+I_2$ is an ideal of $A$,can we conclude that $I_2\subset I_1$?
My thought:if there exists an element $x\in I_2$ which is not in $I_1$,then $I_1+I_2$ properly contains $I_1$,but $I_1$ is maximal,we get a contradiction.
Is my thought correct?
If you add the additional hypothesis that $I_1+I_2$ is a proper ideal of $A$, then you can conclude that $I_2\subset I_1$. For in this case we have $I_1=I_1+I_2\supset I_2$. But it may happen that $I_1+I_2=A$, in which case you cannot conclude that $I_2\subset I_1$.
Note that this has nothing to do with $A$ being a $C^*$-algebra. Your question could be stated for $A$ an arbitrary ring, and the same conclusion holds.