Definition 10.131.1. Let $\varphi : R \to S$ be a ring map and let $M$ be an $S$-module. A derivation, or more precisely an $R$-derivation into $M$ is a map $D : S \to M$ which is additive, annihilates elements of $\varphi (R)$, and satisfies the Leibniz rule: $D(ab) = aD(b) + bD(a)$.

Comment #1849 by Peter Johnson on

Maybe too fussy, but since you implicitly use left modules, should say something somewhere about bimodules/right action to explain D(a)b.

A matter of taste, but I would prefer to see something like "R-algebra" rather than $\varphi: R \to S$, with "R-linear" put in the def. rather than "additive, annihilates elements of $\varphi(R)$", giving hint to calculate D(1) = D(1.1). [Even without assuming the fussier detail of unitary action 1m=m, still get 1D(1)=0, then D(1)=0, then D(r.1) = 0.]

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