You are here
Homereflexive module
Primary tabs
reflexive module
Let $R$ be a ring, and $M$ a right $R$module. Then its dual, $M^{*}$, is given by $\hom(M,R)$, 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 $\hat{m}$, since it only depends on $m$. For any $m\in M$, the mapping
$m\mapsto\hat{m}$ 
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

any free module is torsionless.

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

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

$R$ is reflexive.

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

based on the two immediately preceding properties, any finitely generated projective module is reflexive.
Mathematics Subject Classification
16D90 no label found16D80 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections