Hi Tsiao, when you do the Trim it slices the skin of your solid up into separate pieces and so your objects after that are no longer closed solids that enclose a volume.
The boolean commands are mostly oriented towards working on solids, they figure out which pieces to keep based on which solid volume the pieces are in. Once your object is no longer a solid and just an open surface you will get the kind of results that you show there if you try to do booleans on it.
Once you are working with surfaces instead of solids you will usually need to use Trim to cut things up rather than booleans - Trim is somewhat similar to booleans in that it's job is to slice objects with one another, but with Trim you select which pieces you want to keep or discard manually. You can kind of think of booleans as a sort of high level batch mode operation of Trim that you can use when you have solids, where it does a combination of Trim + selecting pieces to discard based on which volume they are contained in + join of the pieces. Since they basically combine several actions together they are convenient to use if you are able to, but when you are working with surfaces you will generally not be able to use them and instead must use Trim.
There are a variety of different ways you could solve your problem though - one is that instead of doing the extrude with "both sides", just do it one side towards the inside of your shape and then delete the outer surface which will leave just the recessed portion and then use Join to glue those together. That's the approach shown in that previous link:
http://moi3d.com/forum/index.php?webtag=MOI&msg=3024.4
Check out the steps shown in that link a bit more closely and you'll see that one of the steps is to delete the front face of the extrude, just leaving the recessed portion of the extrusion.
The other way you could do it is to select your extruded object, and then do a Ctrl+C to copy it to the clipboard, then undo all the way back to before you did the trim in the first place so that you've got the ring as a closed solid instead of an open surface, then paste in your extrusion and now you will have solids and so you can use a boolean at that point.
Let me know if you're still stuck.
- Michael
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