Two problems on submodules of H²(Dⁿ)

Given any shift-invariant closed subspace S (aka submodule) of the Hardy space over the unit polydisc H²(Dⁿ) (where n ≥ 2), let Rzⱼ:=Mzⱼ|S, and Ezⱼ:=PS∘ evzⱼ, for each j ∈ \1,…,n\. Here, evzⱼ is the operator evaluating at $0$ in the zⱼ-th variable. In this article, we prove that given any subset Λ ⊆ \1,…,n\, there exists a collection of one-variable inner functions \φ_λ (z_λ)\λ ∈ Λ on Dⁿ, such that \[ S = ∑λ ∈ Λ φ_λ (z_λ)H^2(D^n), \] if and only if the conditions (IS-EzₖEzₖ^*)(IS-RzₖRzₖ^*)=0 for all k ∈ \1,,n\ Λ, and (IS-E_zᵢE_zᵢ^*)(IS-R_zᵢR_zᵢ^*)(IS-E_zⱼE_zⱼ^*)(IS-R_zⱼR_zⱼ^*)=0 for all distinct i,j ∈ Λ are satisfied. Following this, we study R.G. Douglas's question on the commutativity of orthogonal projections onto the corresponding quotient modules.