  Using a new definition of generalized divisors we prove that the lattice of
such divisors for a given linear partial differential operator is modular and
obtain analogues of the well-known theorems of the Loewy-Ore theory of
factorization of linear ordinary differential operators. Possible applications
to factorized Groebner bases computations in the commutative and
non-commutative cases are discussed, an application to finding criterions of
Darboux integrability of nonlinear PDEs is given.
