The Stacks Project proves Thomason's insight that
compact objects of the derived category $\simeq$ bounded complexes of finitely generated projective modules
in Section 15.78, but the running conventions of the Stacks Project assume that all rings are commutative.
I believe the insight is originally from the Thomason-Trobaugh paper in the Grothendieck Festschrift, but this paper also works exclusively with commutative rings (and schemes).
What is the situation for non-commutative rings?