In this paper we consider the problem of inverting an n×n]]> circulant matrix with entries over Z_m]]>. We show that the algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over Z_m]]>. We present three different algorithms which do not use this approach. Our algorithms require different degrees of knowledge of m and n, and their costs range -- roughly -- from nlognloglogn]]> to n log²nloglogn logm]]> operations over Z_m]]>. We also present an algorithm for the inversion of finitely generated bi-infinite Toeplitz matrices. The problems considered in this paper have applications to the theory oflinear Cellular Automata.