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In mathematics, the dual module of a left module M over a ring R is the set of module homomorphisms from M to R with the pointwise right module structure. The dual module is typically denoted M or HomR.
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. 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 = Spec {\displaystyle G=\operatorname {Spec} } is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring k, then the Cartier dual G D {\displaystyle G^{D}} is the Spec of the dual k-module of A.
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