 Dual module

For a left module $M$ over a ring $R$, its dual module is $M^{\ast }={\mathrm{Hom}}_R(M,R)$. It is naturally a right module, withRight multiplication preserves left linearity: $f(am)r=af(m)r$. If $R$ is commutative, this can equally be regarded as a left $R$-module, with the usual pointwise scalar action.

For a finite free module with a specified basis, its coordinate functionals form a dual basis. Over $\mathbb{Z}$, the dual of $\mathbb{Z}/n\mathbb{Z}$ is zero for $n>1$, since an additive map from a finite group to the torsion-free group $\mathbb{Z}$ is zero. Duality therefore need not recover every module.

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.