For a left module over a ring , its dual module is . It is naturally a right module, withRight multiplication preserves left linearity: . If is commutative, this can equally be regarded as a left -module, with the usual pointwise scalar action.
For a finite free module with a specified basis, its coordinate functionals form a dual basis. Over , the dual of is zero for , since an additive map from a finite group to the torsion-free group is zero. Duality therefore need not recover every module.
The double dual module is obtained by dualizing the dual module again. For a left module over a possibly noncommutative ring , the correct sided formula isIt is again a left module. The evaluation homomorphism sends to the functional ; right linearity of this functional follows from .
Over a commutative ring, both Hom operations can be written with subscript . Evaluation is an isomorphism for finite projective modules, while a reflexive module is defined by requiring it to be an isomorphism. It need not even be injective in general: the double dual of is zero when .
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The term "dual module" can have different meanings depending on the context in which it is used, particularly in fields like electronics, education, and software. Here are a few interpretations: 1. **Electronics**: In the context of electronics, a dual module may refer to a component that contains two functional units in a single package.