Dual module - Knowlpedia

Let $R$ be a commutative ring and $M$ an $R$ -module. The dual module of $M$ is

M^\vee := \mathrm{Hom}_R(M,R),

with scalar multiplication $(r\varphi)(m)=r\varphi(m)$ .

Duality is contravariant and interacts tightly with tensors via the tensor–Hom adjunction; it packages bilinear pairings $M\times N\to R$ as linear maps $M\to N^\vee$ . For free modules of finite rank, duality is well-behaved and compatible with the notion of a basis, producing a dual basis.

Examples

If $M\cong R^n$ is free with basis $e_1,\dots,e_n$ , then $M^\vee\cong R^n$ with dual basis $e_1^\vee,\dots,e_n^\vee$ characterized by $e_i^\vee(e_j)=\delta_{ij}$ .
If $V$ is a finite-dimensional vector space over a field, then $V^\vee$ is the usual linear dual space.