Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a fully faithful, exact embedding into the module category of some ring.
Therefore, I'd like a theorem like this: Any small triangulated category is a fully faithful, triangulated subcategory of the unbounded derived category of modules on some ring.
My guess is that this fails to be true, for similar reasons to a triangulated category not always being the derived category of its core. Is there a simple example of this? -and- Is there a set of properties that do imply the above theorem?