dual module

In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure.{{cite book|title=Algebra I|author=Nicolas Bourbaki|authorlink = Nicolas Bourbaki|date=1974|publisher=Springer|isbn=9783540193739}}{{cite book|title=Algebra|author=Serge Lang|authorlink = Serge Lang|date=2002|publisher=Springer|isbn=978-0387953854}} The dual module is typically denoted M^∗ or {{nowrap|Hom_R(M, R)}}.

If the base ring R is a field, then a dual module is a dual vector space.

Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective.

Example: If $G\; =\; \backslash operatorname\{Spec\}(A)$ is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring R, then the Cartier dual $G^D$ is the Spec of the dual R-module of A.