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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?

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It's the same.

First, you can notice that if you have a set of compact objects $F$ in a triangulated category $S$ such that there's no proper triangulated, coproduct-closed subcategory containing $F$, the minimal thick subcategory containing $F$ is exactly the subcategory of compact objects. (It's more or less a consequence of the triangulated Brown representability theorem.)

Then if $T$ is $D(R{\operatorname{-mod}})$, $R$ is a compact object in it (since homs from it are zeroth cohomology); the smallest thick subcategory containing $R$ is the subcategory of perfect complexes, so by the observation above it is the subcategory of compact objects.

If you are not satisfied with my brief explanation, this is written more carefully in Neeman's book on triangulated categories.

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