Injective maps on primitive sequences over Z/(pe)

Let Z(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence Z/(pe), there is a unique p-adic decomposition a = a0 + a1 · p + ⋯ + ae-1 · pe-1, where each ai can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and Gd (f(x), pe the set of all primitive sequences generated by f(x) over Z/(pe). For μ(χ) ∈ Z/(p)[x] with deg(μ(χ)) ≥ 2 and ged(1 + deg(μ(χx)), p - 1) = 1 set. φe-1(χ0, χ1 ⋯, χe-1) = χe-1 · [μ(χe-2) + ηe-3(χ0, χ1, ⋯, χe-3)] + ηe-2(χ0, χ1, ⋯, χe-2), which is a function of e variables over Z/(p). Then the compressing map φe-1 : G'(f(χ), pe) → (Z/(p))infin;, a → φe-1(a0, a1, ⋯, ae-1) is injective. That is, for a, b ∈ G'(f(χ), pe), a = b if and only if φe-1(a0, a1, ⋯, ae-1) = ?φe-1 (b0, b1, ⋯, be-1). As for the case of e = 2, similar result is also given. Furthermore, if functions φe-1 and ψe-1 over Z/(p) are both of the above form and satisfy φe-1(a0, a1, ⋯, ae-1) = ψe-1(b0, b1, ⋯, be-1) for a, b ∈ G'(f(χ), pe), the relations between a and b, φe-1 and ψe-1 are discussed. © Editorial Committee of Applied Mathematics 2007.