  We define very proper intersections of modules and projective subschemes. It
turns out that equidimensional locally Cohen-Macaulay modules intersect very
properly if and only if they intersect properly. We prove a Bezout theorem for
modules which meet very properly. Furthermore, we show for equidimensional
subschemes $X$ and $Y$: If they intersect properly in an arithmetically
Cohen-Macaulay subscheme of positive dimension then $X$ and $Y$ are
arithmetically Cohen-Macaulay. The module version of this result implies
splitting criteria for reflexive sheaves.
