PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Very high
dual module (Definition)

Let $ R$ be a ring and $ M$ be a left $ R$-module. The dual module of $ M$ is the right $ R$-module consisting of all module homomorphisms from $ M$ into $ R$.

It is denoted by $ M^\ast$. The elements of $ M^\ast$ are called linear functionals.

The action of $ R$ on $ M^\ast$ is given by $ (fr)(m) = (f(m))r$ for $ f \in M^\ast$, $ m \in M$, and $ r \in R$.

If $ R$ is commutative, then every $ R$-module $ M$ is an $ (R,R)$-bimodule with $ rm = mr$ for all $ r \in R$ and $ m \in M$. Hence, it makes sense to ask whether $ M$ and $ M^\ast$ are isomorphic. Suppose that $ b: M \times M \to R$ is a bilinear form. Then it is easy to check that for a fixed $ m \in M$, the function $ b(m, -): M \to R$ is a module homomorphism, so is an element of $ M^\ast$. Then we have a module homomorphism from $ M$ to $ M^\ast$ given by $ m \mapsto b(m,-)$.



"dual module" is owned by Mathprof.
(view preamble)

View style:

See Also: unimodular

Also defines:  linear functional
Log in to rate this entry.
(view current ratings)

Cross-references: function, fixed, bilinear form, isomorphic, commutative, action, homomorphisms, module, right, ring
There are 12 references to this entry.

This is version 7 of dual module, born on 2006-06-14, modified 2006-10-24.
Object id is 8037, canonical name is DualModule.
Accessed 1686 times total.

Classification:
AMS MSC: 16-00 (Associative rings and algebras :: General reference works )
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)