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I've been trying to read some papers on differential graded modules (for example, Keller, Deriving DG categories) In most of literature I found about dg-modules, they define them as right modules (Of course there might be more literature which uses left modules that I didn't find), and I can't find a reason to do so. My one guess is that the right module definition seems more analogous to the definition of sheaves as it is defined by contravariant functors, but I don't know whether such analogy is meaningful or not.

-Is there any historical reason for that? -Is there any application that right module structure arise more naturally?

Thank you!

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    $\begingroup$ As with the Yoneda embedding of the category into (contravariant) presheaves, there is an embedding of $A$ into right $A$-modules. Alternately, there is an embedding of $A^{op}$ into left $A$-modules: you can choose where to put the contravariance, but it has to end up somewhere. . . So that's one reason, but there may be others. $\endgroup$
    – Ben Wieland
    Commented Apr 8, 2016 at 23:13

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Well, first of all, this is perhaps really a matter of taste. But there is good taste :) More seriously, the following argument in favour of right modules is not limited to the DG situation: if you consider $R^n$ for a noncommutative ring $R$, then you perhaps want the matrices $M_n(R)$ to act from the left via usual matrix multiplication. This seems to be reasonable, right? But then $R^n$ becomes a $(M_n(R), R)$-bimodule only if you use the right $R$-module structure. Perhaps one should already do this for vector spaces...

The bimodule aspect becomes relevant whenever you are interested in say Morita theory or similar things.

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