Let Z/(p e ) be the integer residue ring modulo p e with p an odd prime and integer e ≥ 3. For a sequence a over Z/(p e ), there is a unique p-adic decomposition a = a 0 +a 1 ·p + · · ·+ a e−1 · p e−1 , where each a i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e − 1. 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 ). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 + deg(μ(x)), p − 1) = 1, setwhich is a function of e variables over Z/(p). Then the compressing map. 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(a 0 , a 1 , · · · , a e−1 ) = ψe−1(b 0 , b 1 , · · · , b e−1 ) for a, b ∈ G (f (x), p e ), the relations between a and b, ϕe−1 and ψe−1 are discussed. §1 Introduction Let p be a prime and Z/(p e ) the integer residue ring modulo p e , which can also be represented as {0, 1, · · · , p e − 1}. In this correspondence, given a positive integer m ≥ 2, we always consider a(mod m) as an element in {0, 1, · · · , m − 1}.Let f (x) = x n + c n−1 x n−1 + · · · + c 1 x + c 0 be a monic polynomial with degree n ≥ 1 over Z/(p e ). A sequence a = (a(t)) t≥0 over Z/(p e ) satisfying the recursion a(i + n) = −(c 0 · a(i) + c 1 · a(i + 1) + · · · + c n−1 · a(i + n − 1)), i = 0, 1, · · · (1.1)
