Sue,
The other posters touched on this already. I'll just give you a bit more detail on it. Upper torso clothing cosists of several layers: Skin, Undershirt, Shirt, Jacket. SL will bake all 4 layers into what you see as your AV. Jewelry can be a texture applied to one of those layers... OR... it can be a prim attachment. If it's applied to the skin layer, then it becomes a permanent part of the skin layer (like a tattoo) which clothing can't remove. But who wants to wear the same necklace all the time.

If it's a prim necklace, as Joesph pointed out, the chest and spine are the 2 best attachment points, with chest being the normal one. Stomach is slightly less good, and requires some editing to manipulate the prims to the neck area. Left & Right shoulders can also be used, but are worse (as the body turns, the item will disappear).
What you described, as CJ pointed out, you are wearing a clothing layer, rather than a prim. Both CJ & Raindrop gave you good info. The one thing hinted at is that many clothing makers already know about this issue, and willing to work with you if you IM them (not all, mind you). If you are buying a corset that is an undershirt layer and you want to wear a piece of jewelry. IM the 2 makers. Perhaps the jewelry maker can put the jewelry texture on a shirt layer... or maybe the corset maker can.
A general rule of thumb to help you identify what you are looking at in your inventory:
An undershirt layer icon looks like a tank top shirt
A Shirt layer icon looks like a blouse
A Jacket layer icon looks like a dark windbreaker jacket
A prim attachment icon will look like a box (so will an unpacked outfit)
Jackets always cover the other 2 layers
Shirts always cover Undershirt layer
Prims always appear outside clothing layers
Again, this is just generalizations, you will see nuances where a clothing maker will use 2 or more layers to achieve a certain look, like using the jacket layer to add a sash.