As is known (see Kadison-Ringrose, 3.4.1) each closed ideal $I$ in the $C^*$-algebra $C(X)$ of continuous functions on a compact space $X$ has the form $$ I=\{f\in C(X): \ \forall x\in S\quad f(x)=0 \} $$ for some closed subset $S$ in $X$.

Is the same true for the two-sided ideals in the $C^*$-algebra $C(X,M_n)$ of continuous maps $f:X\to M_n$ into the algebra $M_n$ of all $n\times n$ complex matrices?

Morita-Rieffel equivalentand this implies that they have the same lattice of ideals. $\endgroup$