find all ideals of direct sum of simple $C^*$ algebras

Question

Suppose $A=\bigoplus A_n$ where each $A_n$ is a simple $C^*$ algebra(The direct sum is $c_0$ direct sum).I guess all the ideals of $A$ are precisely the direct sum of ideals of $A_n$ .It is easy that the direct sum of ideals of $A_n$ is an ideal of $A$.How to show that all ideas of $A$ is the form of direct sum of ideals of $A_n$?

Answer 1

Take an ideal $J\subset A$. Write $p_n$ for the central projection $(0,\ldots,0,1,0,\dots)$. Then $p_nJ$ is an ideal, and its only nonzero elements (if any) are in the $n^{\rm th}$ component, call this $J_n$. This $J_n$ is an ideal in $A_n$. Now you can check that $J=\bigoplus J_n$. So all ideals in $A$ are direct sums of ideals.