Ideals of inductive limit of $C^{\ast}-$ algebra.

Question

I am trying to understand ideals of direct limits in the category of $C^{\ast}$-algebras.

Let $(A_n,f_n)$ be a direct sequence of $C^{\ast}$-algebras and let $I$ be an ideal of direct limit $\varinjlim A_n $. Is it true that there exists ideals $I_i$ of $A_i$ such that $I= \varinjlim I_i$

Unfortunately I could not find any reference.Any reference or ideas?

Answer 1

A reference could be Lemma III.4.1 in Davidson's C$^*$-algebras. You can simply take $I_j=I\cap A_j$.