TITLE: Freyd-Mitchell for triangulated categories?
QUESTION [16 upvotes]: 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?

REPLY [8 votes]: Don't know if this is germane, but Freyd actually proved that every small triangulated category has an abelian envelope.  That is, you can embed a triangulated category into an abelian category, in fact a Frobenius category, where the triangulated category is the category of projectives=injectives.  I believe he introduced this construction in the same paper, in 1965 or so, in some La Jolla conference proceedings, in which he introduced the generating hypothesis in stable homotopy theory.  It also appears in Amnon Neeman's book on triangulated categories, I am pretty sure.  
Mark