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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 commutativealgebra $\mathscr{S}_1(M,\mathcal{N})$ 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 splittingmap for $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho}$. We prove the splittingmap for $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho}$ is injective if thereexists at least one component of $\mathcal{N}$ such that this component and theboundary 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 $M$ byremoving $k$ open three dimensional balls and adding one marking to each newlycreated sphere boundary component. We prove$\text{dim}_{\mathbb{C}}\mathscr{S}_{q^{1/2}}( M_{k})_{\rho} = 1$ for anymaximal ideal $\rho$ of $\mathscr{S}_1(M_k)$.

Representation reduced stated skein modules and algebras