dualmodule

A dual module, in algebra, is a standard construction taking a module to a space of linear functionals. For a ring R and a left R-module M, the dual module M* is defined as Hom_R(M, R), the set of all R-linear maps from M to R. If R is commutative, M* naturally carries an R-module structure consistent with M; more generally, the dual of a left module is a right module, and the dual of a right module is a left module, with the module actions given by (f·r)(m) = f(m)r or similar conventions depending on left/right orientation.

The dual captures linear functionals on M. For a free module M ≅ R^n, M* ≅ R^n with the dual basis corresponding to the standard coordinate functionals. Over a field K, if M is a vector space, M* is the space of linear functionals on M, and its dimension equals that of M. This yields the familiar dual space of a vector space, commonly used in linear algebra.

Key properties include functorial behavior: a module homomorphism f: M → N induces a dual map f*: N* → M* defined by f*(g) = g ∘ f. This duality is contravariant, reversing arrows. The dual operation is compatible with tensor and Hom constructions in many contexts.

A central theme in duality theory is reflexivity. There is a natural evaluation map ev_M: M → M** (the dual of M*), defined by ev_M(m)(f) = f(m). If M is finitely generated and projective, or more generally finitely generated over a field, ev_M is an isomorphism, so M is naturally identified with its double dual. In general, M** can be larger, and modules may fail to be reflexive.

Dual modules appear across algebra and functional analysis, including representation theory, module theory over rings, and the study of linear functionals in linear algebra.