Compression mappings on primitive sequences over Z/(pe)

Let Z/(p^e) be the integer residue ring with odd prime p≥5 and integer e≥2. For a sequence a_ over Z/(p^e), there is a unique p-adic expansion a_=a_₀+a_·p+...+a__e-1·p^e-1, where each a__i is a sequence over {0,1,...,p-1}, and can be regarded as a sequence over the finite field GF(p) naturally. Let f(x) be a primitive polynomial over Z/(p^e), and G´(f(x),p^e) the set of all primitive sequences generated by f(x) over Z/(p^e). Set φ_e-1 (x₀,...,x_e-1) = x_e-1^k + η_e-2,1(x₀, x₁,...,x_e-2) ψ_e-1(x₀,...,x_e-1) = x_e-1^k + η_e-2,2(x₀,x₁,...,x_e-2) where η_e-2,1 and η_e-2,2 are arbitrary functions of e-1 variables over GF(p) and 2≤k≤p-1. Then the compression mapping φ_e-1:{G´(f(x),p^e) → GF(p)^∞ a_ → φ_e-1(a_₀,...,a__e-1) is injective, that is, a_ = b_ if and only if φ_e-1(a_₀,...,a__e-1) = φ_e-1(b_₀,...,b__e-1) for a_,b_ ∈ G´(f(x),p^e). Furthermore, if f(x) is a strongly primitive polynomial over Z/(p^e), then φ_e-1(a_₀,...,a__e-1) = ψ_e-1(b_₀,...,b__e-1) if and only if a_ = b_ and φ_e-1(x₀,...,x_e-1) = ψ_e-1(x₀,...,x_e-1) for a_,b_ ∈ G´(f(x),p^e).