The module isomorphism problem reconsidered

This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a Magma implementation of our algorithm.