The Module Isomorphism Problem for Finite Rings and Related Results

TL;DR

This paper introduces two efficient algorithms to determine module isomorphism over finite rings, improving existing methods by avoiding certain complex computations and broadening applicability.

Contribution

The paper presents two deterministic polynomial-time algorithms for module isomorphism over finite rings, without requiring the ring to contain a field or computing the Jacobson radical.

Findings

Algorithms decide isomorphism in polynomial time

Able to find largest common direct summand

Determine minimum number of generators

Abstract

Let $R$ be a finite ring and let $M, N$ be two finite left $R$-modules. We present two distinct deterministic algorithms that decide in polynomial time whether or not $M$ and $N$ are isomorphic, and if they are, exhibit an isomorphism. As by-products, we are able to determine the largest isomorphic common direct summand between two modules and the minimum number of generators of a module. By not requiring $R$ to contain a field, avoiding computation of the Jacobson radical and not distinguishing between large and small characteristic, both algorithms constitute improvements to known results.