  In this note we study the logarithmic derivation module of a non-free
arrangement. We prove a generalized addition theorem for all arrangements. This
addition theorem allows us to find various relationships between non-free
arrangements, free arrangements and restriction counts. For graphic
arrangements we can use these results to find a lower bound for the maximal
degree generator in terms of triangles in the associated graph. We also apply
these results to the case of hypersolvable arrangements where we define
hyperexponents and use them to find a lower bound for their maximal degree
generator.
