1. Introduction

This paper is a continuation of a joint paper with B. Martin [MS] dealing with the problem of direct sum decompositions. The techniques of that paper are used to decide wether two modules are isomorphic or not. An positive answer to this question has many applications - for example for the classification of maximal Cohen-Macaulay module over local algebras as well as for the study of projective modules. Up to now computer algebra is normally dealing with equality of ideals or modules which depends on chosen embeddings. The present algorithm allows to switch to isomorphism classes which is more natural in the sense of commutative algebra and algebraic geometry.

Let R be a finitely generated (local) k-algebra without zerodivisors. Let M and M' be two modules given via minimal representation matrices A and A'. Then $M\simeq M'$ if and only if there are matrices $U,V\in Gl(R)$ such that UAV=A'. We shall descibe a finite algorithm to either compute the matrices U and V or to disprove isomorphism.