Hi Rudy, I noticed that the ends of your ring "B" are open - this means that that shape is only a surface and not a solid that encloses a volume. It is possible to do booleans using surfaces, but in some situations booleans can function better with solids, because a solid will always cut things into 2 pieces.
A surface doesn't always cut something into distinct pieces, for example here is a solid box with a surface:
The box is not able to be cut using a boolean by that surface, because the surface does not slice the box into 2 fully separate pieces, it just cuts a kind of infinitely thin slit in it. Booleans only work with fully dividing cuts.
Basically, your open surface was a similar case to this.
You can turn your ring "B" into a solid by selecting it and then running Construct / Planar - that put "end caps" on the open ends of that shape, sealing it into a solid. Planar can be used to seal off planar openings in a surface to turn it into a solid, or it also can create a single planar surface shape from a set of planar curves.
Anyway, after making ring "B" into a solid, I was able to position those pieces over top of each other and do a boolean difference, the result is this:
Is that the type of shape you're looking for? Do you want that inner channel to sort of stop near the bottom like that?
One other tip in general for booleans - when you create a piece that you want to use as a cutting piece in a boolean, it can be sometimes better if you create the piece such that it punches through the other object, rather than having areas that are exactly (or more problematic, only nearly exactly) overlapping. Like for instance in this case it would be not a bad idea to make the inside radius of your "B" cutting piece to be a smaller radius instead of the same as the outside piece. In this case it actually works fine though. But sometimes slight differences between overlapping surfaces can create problems when MoI tries to calculate intersections between the 2 objects.
- Michael
