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 areused to decide wether two modules are isomorphic or not. An positive answer to this question has many applications - for example for the classification ofmaximal Cohen-Macaulay module over local algebras as well as for the study of projective modules. Up to now computer algebra is normally dealing withequality of ideals or modules which depends on chosen embeddings. The present algorithm allows to switch to isomorphism classes which is more natural inthe sense of commutative algebra and algebraic geometry.
A new criteria for indecomposability of vector bundles on projective varieties is presented. It is deduced from a new finite algorithm computing direct sumdecompositions of graded modules over graded algebras. This algorithm applies as well to modules over local complete algebras over a field.