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Abstract Algebra

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Dual of a Module over a Ring: Definition and Structure

ABSALG-ZRMLJ1

Let $R$ be a ring with identity. Let $M$ be a left unital $R$-module and $N$ a right unital $R$-module.

Denote by $M^\ast$ the left $R$-module homomorphisms from $M$ to $R$ and by $N^\ast$ the right $R$-module homomorphisms from $N$ to $R$. These are called the dual modules to $M$ and $N$, respectively.

Which of the following are always correct?

Select ALL that apply.

A

If $M$ is free with finite basis, then $(M^\ast)^\ast$ is a left $R$-module isomorphic to $M$.

B

If $R$ is commutative, then $M^\ast$ is an $R$-module isomorphic to ${\rm Hom}_R(R,M)$.

C

If $R$ is a field, then $M^\ast$ is an $R$-vector space isomorphic to $M$.

D

If $R$ is a field, then $(M^\ast)^\ast$ is an $R$-vector space isomorphic to $M$.

E

$M^\ast$ is a right $R$-module.