Generalized Loewy-decomposition of d-modules

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.