$\oplus_{i\in I}A_i$ denotes the $c_0$ direct sum of $C^*$ algebras $A_i$, $\prod_{i\in I}A_i$ is the $l^\infty$ direct sum of $A_i$.We know that $\oplus_{i\in I}A_i$ is the essential ideal of $\prod_{i\in I}A_i$.
Do there exist other proper ideals of $\prod_{i\in I}A_i$ which contain $\oplus_{i\in I}A_i$? If there exist,how to construct these ideals?
Yes. For any non-principal ultrafilter $U$ on $I$, you can consider the ideal of elements of $\prod A_i$ which are equal to zero $U$-almost everywhere. The quotient of $\prod A_i$ by this ideal is the corresponding ultraproduct.