Discrete spectra of $C^{*}$-algebras and complemented submodules in Hilbert $C^{*}$-modules

Let $A$ be a $C^{*}$-algebra and let $X$ be a full (right) Hilbert $A$-module. It is shown that if the spectrum $\widehat A$ of $A$ is discrete, then every closed $\mathcal{K}(X)$-$A$-submodule of $X$ is complemented in $X$, and conversely that if $\widehat A$ is a $T_{1}$-space and if every closed $\mathcal{K}(X)$-$A$-submodule of $X$ is complemented in $X$, then $\widehat A$ is discrete.