We study Van den Bergh's non-commutative symmetric algebra 𝕊nc(M) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show that 𝕊nc(M) is coherent, and its proj category ℙnc(M) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ℙnc(M) is hereditary and there is a structure theorem for sheaves on ℙnc(M) analogous to that for ℙ1.

In this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Motlocdg in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Motlocdg(e) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a precise connection between Toën's secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild homology.