  In this paper, we describe primitive ideal space of the $C^*$-algebra
$C^*(\Lambda)$ associated to any locally convex row-finite $k$-graph $\Lambda$.
To do this, we will apply the Farthing's desourcifying method on a recent
result of Carlsen, Kang, Shotwell, and Sims. We also characterize certain
maximal ideals of $C^*(\Lambda)$.
  Furthermore, we study the decomposability of $C^*(\Lambda)$. We apply the
description of primitive ideals to show that if $I$ is a direct summand of
$C^*(\Lambda)$, then it is gauge-invariant and isomorphic to a certain
$k$-graph $C^*$-algebra. So, we may characterize decomposable higher-rank
$C^*$-algebras by giving necessary and sufficient conditions for the underlying
$k$-graphs. Moreover, we determine all such $C^*$-algebras which can be
decomposed into a direct sum of finitely many indecomposable $C^*$-algebras.
