Encyclopedia

Dual module

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated June 5, 2025

In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure. The dual module is typically denoted M^∗ or Hom_R(M, R).

If the base ring R is a field, then a dual module is a dual vector space.

Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective.

Example: If $G=\operatorname {Spec} (A)$ is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring R, then the Cartier dual $G^{D}$ is the Spec of the dual R-module of A.

Watch videos about Dual moduleExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Dual module”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Dual module

Sources and credits

Related topics

Sheaf of modules