Torsion 0-cycles with modulus on affine varieties

In this note, we show that given a smooth affine variety X over an algebraically closed field k and an effective (possibly non-reduced) Cartier divisor on it, the Chow group of zero cycles with modulus CH0(X|D) is torsion free, except possibly for p-torsion if the characteristic of k is p>0. This generalizes to the relative setting classical results of Rojtman (for X smooth) and of Levine (for X singular).