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Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.

Explicitly how would one go about computing exactly what the module of $R$-differentials from $R<x,y>$ to itself?

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    $\begingroup$ The module of $R$-derivations is just $R<x,y>^2$: given $P,Q$ in $R<x,y>$, there is one and only one derivation $D$ such that $Dx=P$ and $Dy=Q$. See Bourbaki, Algebra III, §10, Proposition 14. $\endgroup$
    – abx
    Commented Apr 11, 2014 at 5:56
  • $\begingroup$ Perfect, I'll look it up and post the answer to this question, thanks ABx $\endgroup$
    – ABIM
    Commented Apr 11, 2014 at 18:25
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    $\begingroup$ This is not really a question appropriate for this site. The free álgebra has a universal property with respect to derivations which makes the answer obvious. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 18, 2014 at 19:47

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