Commutative Ring Theory/Derivations

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Proposition (alternative construction of the universal derivation):

Let be a unital -algebra. Note that becomes an -module via the linear extension of the operation . We then have a morphism of -modules

,

where the dot indicates the algebra multiplication of . Set and . Then

is a derivation, and we have an isomorphism inducing a commutative diagram

Isomorphism of universal derivations.svg

Proof: Note first that is a derivation. This takes some explaining. First, note that for arbitrary the element is in . Moreover, from this follows that the element

is in for arbitrary.

Hence, from the universal property of , we obtain a unique morphism of -modules that makes the diagram

Isomorphism of universal derivations.svg

commutative. We construct an inverse map to . Namely, on we can define the map