General Ring Theory/Derivations

derivation Let be a unital -algebra. Moreover, let be a module over the algebra . Then a derivation on over with values in is a morphism of -modules such that for all . We denote the -module of all derivations of over with values in by . universal derivation Let be a unital -algebra. Then the universal derivation of over is the derivation , where is the -module that arises by taking the free -module generated by the formal symbols (one for each ) and dividing out the ideal generated by elements of the form as well as (where , ), and sends to . universal property of the universal derivation Let be a unital -algebra and let be a module over the algebra . Then for each , there exists a unique morphism of -modules such that . This pairing induces an isomorphism of -modules . For uniqueness, note that the universal derivation is surjective, so is determined by . For existence, note that if is the free -module with generators for all , then there is a morphism that sends to , and it factors over the quotient in the definition of by the properties of . universal derivation is unique The universal derivation of a unital -algebra together with its target module is uniquely defined. This follows immediately from the universal property that defines the corresponding object uniquely.