The module isomorphism problem reconsidered

Abstract

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and Magma, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures -such as groups, and almost any type of algebra -seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem? 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.

Linear Representation of Graphs: Applications to Molecular Graphs

“…Proof. The proof follows from the fact that isomorphism problem of Rmodules is done in polynomial time, [3][4][5]12].…”

Linear representation of graphs: applications to molecular graphs

“…Proof The proof follows from the fact that isomorphism problem of R-modules is done in polynomial time, [2][3][4]10].…”

The isomorphism problem for uniserial modules over an arbitrary ring

“…We should also note that in 2006, it is proven by Příhoda that for two uniserial modules U and V over any ring R, U ∼ = V if and only if there is a monomorphism f : U → V and an epimorphism g : U → V ([13, Remark 2.1]). We refer to [5] for very interesting and recent results on computer algebra concerning the so-called "Module Isomorphism Problem" and many related isomorphism problems for algebraic structures. Note that there exists an artin algebra having two nonisomorphic uniserial left modules of length two with the same socle and top.…”