Relative determinant of a bilinear module

The aim of the paper is to generalize the (ultra-classical) notion of the determinant of a bilinear form to the class of bilinear forms on projective modules without assuming that the determinant bundle of the module is free. Successively it is proved that this new definition preserves the basic properties, one expects from the determinant. As an example application, it is shown that the introduced tools can be used to significantly simplify the proof of a recent result by B. Rothkegel.