Note on compact operators on a Hilbert C⁎-module

In this note, we characterize the tracial topological rank of the ideal of compact operators on a Hilbert A-module. It is also proved that this ideal preserves exactness, quasidiagonality, and nuclearity if the C⁎-algebra A has these properties, respectively. Finally, a necessary and sufficient condition is given for this ideal K(E) being an AT-algebra when E is a Hilbert A-module over an AT-algebra A.