Dual module

In mathematics, the dual module of a left (resp. right) module M over a ring R is the set of module homomorphisms from M to R with the pointwise right (resp. left) module structure. The dual module is typically denoted M∗ or HomR(M, R). In mathematics, the dual module of a left (resp. right) module M over a ring R is the set of module homomorphisms from M to R with the pointwise right (resp. left) module structure. The dual module is typically denoted M∗ or HomR(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 = Spec ⁡ ( A ) {displaystyle G=operatorname {Spec} (A)} 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.