Maximal modular ideal

Question

I met with some troubles with the two concepts:maximal ideal and maximal modular ideal in $C^*$ algebras.

If $I$ is a maximal modular ideal in a $C^*$ algebra $A$,does this imply that for any other proper ideal $J$ in $A$ ,$J$ is contained in $I$,or for any other proper modular ideals $J^\prime$ in $A$ we have $J^\prime\subset I$?

Answer 1

Neither of the implications hold. Consider $A=C_0(\mathbb R)$, and $I=\{f\in C_0(\mathbb R)\mid f(0)=0\}$. This is a maximal modular ideal, but it does not contain all other ideals, or all other modular ideals.