Dual module

A dual module of an $R$-module $M$ (for a commutative ring $R$) is

viewed as an $R$-module via the standard structure on the Hom module when $R$ is a commutative ring .

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.