Injectivity of compressing maps on primitive sequences over Z/(pe)

Let Z/(p(e)) be the integer residue ring with odd prime p and integer e >= 2. For a sequence a over Z/(p(e)), one has a unique p-adic expansion (a) under bar = (a) under bar (0) + (a) under bar (e-1) center dot p(e-1), where (a) under bar (i) can be regarded as a sequence over Z/(p) for 0 <= i <= e - 1. Let f(x) be a strongly primitive polynomial over Z/ (p(e)) and. G'(f(x), p(e)) be the set of all primitive sequences generated by f(x) over Z/(p(e)). Recently, the authors, Xuan-Yong Zhu and Wen-Feng Qi, have proved that for a function W phi(x(0),...., x(e-1)) = g(x(e-1)) + n(x(0)...., x(e-2)) over Z / (p) and (a) under bar. (b) under bar is an element of G'(f(x),p(e)), where 2 <= deg g <= p - 1, phi( (a) under bar (0),(a) under bar (1), . . . , alpha(e-1)) phi((b) under bar (0), (b) under bar (1), . . . ,(b) under bar (e-1)) if and only if (a) under bar = (b) under bar. To further complete their work, we show that such injectivity also holds for deg g = 1. That is for a function phi(x(0,) . . . , x(e-1)) = x(e-1) + eta(x(0), . . . , x(e-2)) over Z/(p) and (a) under bar, (b) under bar is an element of G'(f(x),p(e)), phi((a) under bar (0),(a) under bar (1), . . , alpha(e-1)) = phi((b) under bar (0),(b) under bar1, . . . , (b) under bar (e-1)) if and only if (a) under bar = (b) under bar.