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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,-)$.



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See Also: unimodular

Also defines:  linear functional
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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 MSC16-00 (Associative rings and algebras :: General reference works )

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