Testing isomorphism of modules

We present a new deterministic algorithm to test constructively for isomorphism between two given finite-dimensional modules of a finitely generated algebra. The algorithm uses only basic field operations; for arbitrary fields, this is not possible with the existing methodology. Furthermore, the number of field operations used by the algorithm is bounded by a polynomial in the length of the input. The algorithm has been implemented in the computer algebra system Magma and we report on its performance. Our approach has applications to other problems concerning decompositions of modules.