A Brian\c{c}on-Skoda type result for a non-reduced analytic space

We present here an analogue of the Brian\c{c}on-Skoda theorem for a germ of an analytic space $Z$ at 0, such that $O_{Z,0}$ is Cohen-Macaulay, but not necessarily reduced. More precisely, we find a sufficient condition for membership of a function in a power of an arbitrary ideal $a^l\subset O_{Z,0}$ in terms of size conditions of Noetherian differential operators applied to that function. This result generalizes a theorem by Huneke in the reduced case.