Dual module

In mathematics, the dual module of a left (respectively right) module "M" over a ring "R" is the set of module homomorphisms from "M" to "R" with the pointwise right (respectively left) module structure. The dual module is typically denoted "M"∗ or .
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 formula_1 is a finite commutative group scheme represented by a Hopf algebra "A" over a commutative ring "k", then the Cartier dual formula_2 is the Spec of the dual "k"-module of "A".