reflexive module
Let $R$ be a ring, and $M$ a right $R$module. Then its dual, ${M}^{*}$, is given by $\mathrm{hom}(M,R)$, and has the structure^{} of a left module over $R$. The dual of that, ${M}^{**}$, is in turn a right $R$module. Fix any $m\in M$. Then for any $f\in {M}^{*}$, the mapping
$$f\mapsto f(m)$$ 
is a left $R$module homomorphism^{} from ${M}^{*}$ to $R$. In other words, the mapping is an element of ${M}^{**}$. We call this mapping $\widehat{m}$, since it only depends on $m$. For any $m\in M$, the mapping
$$m\mapsto \widehat{m}$$ 
is a then a right $R$module homomorphism from $M$ to ${M}^{**}$. Let us call it $\theta $.
Definition. Let $R$, $M$, and $\theta $ be given as above. If $\theta $ is injective^{}, we say that $M$ is torsionless. If $\theta $ is in addition an isomorphism^{}, we say that $M$ is reflexive^{}. A torsionless module is sometimes referred to as being semireflexive.
An obvious example of a reflexive module is any vector space^{} over a field (similarly, a right vector space over a division ring).
Some of the properties of torsionless and reflexive modules are

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any free module^{} is torsionless.

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any direct sum^{} of torsionless modules is torsionless; any submodule^{} of a torsionless module is torsionless.

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based on the two properties above, any projective module^{} is torsionless.

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$R$ is reflexive.

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any finite direct sum of reflexive modules is reflexive; any direct summand^{} of a reflexive module is reflexive.

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based on the two immediately preceding properties, any finitely generated projective module is reflexive.
Title  reflexive module 

Canonical name  ReflexiveModule 
Date of creation  20130322 19:22:38 
Last modified on  20130322 19:22:38 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16D90 
Classification  msc 16D80 
Defines  torsionless 
Defines  reflexive 