  For any marked three manifold $(M,\mathcal{N})$ and any quantum parameter
$q^{\frac{1}{2}}$ (a nonzero complex number), we use
$\mathscr{S}_{q^{1/2}}(M,\mathcal{N})$ to denote the stated skein module of
$(M,\mathcal{N})$.
  When $q^{\frac{1}{2}}$ is a root of unity of odd order, the commutative
algebra $\mathscr{S}_1(M,\mathcal{N})$ acts on
$\mathscr{S}_{q^{1/2}}(M,\mathcal{N})$. For any maximal ideal $\rho$ of
$\mathscr{S}_1(M,\mathcal{N})$, define
$\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho} =
\mathscr{S}_{q^{1/2}}(M,\mathcal{N})\otimes _{\mathscr{S}_1(M,\mathcal{N})}
(\mathscr{S}_1(M,\mathcal{N})/\rho)$. We prove the splitting map for
$\mathscr{S}_{q^{1/2}}(M,\mathcal{N})$ respects the
$\mathscr{S}_1(M,\mathcal{N})$-module structure, so it reduces to the splitting
map for $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho}$. We prove the splitting
map for $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho}$ is injective if there
exists at least one component of $\mathcal{N}$ such that this component and the
boundary of the splitting disk belong to the same component of $\partial M$.
  Let $M$ be a connected closed three manifold. For any positive integer $k$,
we use $M_{k}$ to denote the marked three manifold obtained from $M$ by
removing $k$ open three dimensional balls and adding one marking to each newly
created sphere boundary component. We prove
$\text{dim}_{\mathbb{C}}\mathscr{S}_{q^{1/2}}( M_{k})_{\rho} = 1$ for any
maximal ideal $\rho$ of $\mathscr{S}_1(M_k)$.
