Starting from the well-known factorization of linear ordinary differential equations, we define the generalized Loewy decomposition for a  D -module. To this end, for any module  I , overmodules  J  ⊇  I  are constructed. They subsume the conventional factorization as special cases. Furthermore, the new concept of the module of relative syzygies  Syz ( I,J ) is introduced. The invariance of this module and its solution space w.r.t. the set of generators is shown. We design an algorithm which constructs the Loewy-decomposition for finite-dimensional and some kinds of general  D  modules. These results are applied for solving various second- and third-order linear partial differential equations.
['Dima Grigoriev', 'Fritz Schwarz']
Generalized Loewy-decomposition of d-modules