Let G be a cyclic group of prime power order pk, and let I be the augmentation ideal of the integral group ring Z[G]. We define a derivation on Z/pkZ[G], and show that for 2 ≤ n ≤ p, an element α ∈ I is in In if and only if the i-th derivative of the image of α in Z/pkZ[G] vanishes for 1 ≤ i ≤ (n - 1).
