Commutative Ring Theory/Derivations

General Ring Theory/Derivations 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 File:Isomorphism of universal derivations.svg 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 File:Isomorphism of universal derivations.svg commutative. We construct an inverse map to . Namely, on we can define the map