Injectivity of Compressing Maps on Primitive Sequences Over ${\BBZ}/(p^{e})$

Abstract

Let Zopf/(pe) be the integer residue ring with odd prime p and integer eges2. For a sequence a_ over Zopf/(pe), one has a unique p-adic expansion a_=a_0+a_1.p+...+a_(e-1).pe-1, where a_i can be regarded as a sequence over Zopf/(p) for 0lesilese-1. Let f(x) be a strongly primitive polynomial over Zopf/(pe) and G'(f(x), pe) be the set of all primitive sequences generated f(x) by over Zopf/(pe). Recently, the authors, Xuan-Yong Zhu and Wen-Feng Qi, have proved that for a function phi(x0,...,xe-1)=g(xe-1)+eta(x0,...,xe-2)over Zopf/(p) and a_,b_isinG'f(x),pe), where 2lesdeg glesp-1, phi(a_0,a_1...,a_e-1)=phi(b_0,b_1...,b_e-1) if and only if a_=b_. To further complete their work, we show that such injectivity also holds for deg g=1. That is for a function phi(x0,...,xe-1)=xe-1+eta(x0,...,xe-2)over Zopf/(p) and a_,b_isinG'f(x),pe), phi(a_0,a_1...,a_e-1)=phi(b_0,b_1...,b_e-1) if and only if a_=b_.