Two problems on submodules of $H^2({\mathbb{D}}^n)$

Given any shift-invariant closed subspace $\mathcal{S}$ (aka submodule) of the Hardy space over the unit polydisc $H^2({\mathbb{D}}^n)$ (where $n\ge 2$), let $R_{z_j}:=M_{z_j}{\mathrm{\mid }}_{\mathcal{S}}$, and $E_{z_j}:=P_{\mathcal{S}}\circ e{v}_{z_j}$, for each $j\in \{1,\dots ,n\}$. Here, $e{v}_{z_j}$ is the operator evaluating at $0$ in the $z_j$-th variable. In this article, we prove that given any subset $\mathrm{\Lambda }\subseteq \{1,\dots ,n\}$, there exists a collection of one-variable inner functions $\{{\phi }_{\lambda }(z_{\lambda }){\}}_{\lambda \in \mathrm{\Lambda }}$ on ${\mathbb{D}}^n$, such that $$\mathcal{S}=\sum _{\lambda \in \mathrm{\Lambda }}{\phi }_{\lambda }(z_{\lambda })H^2({\mathbb{D}}^n),$$ if and only if the conditions $(I_{\mathcal{S}}-E_{z_k}{E}_{z_k}^{\ast })(I_{\mathcal{S}}-R_{z_k}{R}_{z_k}^{\ast })=0$ for all $k\in \{1,\dots ,n\}\setminus \mathrm{\Lambda }$, and $(I_{\mathcal{S}}-E_{z_i}{E}_{z_i}^{\ast })(I_{\mathcal{S}}-R_{z_i}{R}_{z_i}^{\ast })(I_{\mathcal{S}}-E_{z_j}{E}_{z_j}^{\ast })(I_{\mathcal{S}}-R_{z_j}{R}_{z_j}^{\ast })=0$ for all distinct $i,j\in \mathrm{\Lambda }$ are satisfied. Following this, we study R.G. Douglas’s question on the commutativity of orthogonal projections onto the corresponding quotient modules.