# 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 semi-reflexive.

Some of the properties of torsionless and reflexive modules are

Title reflexive module ReflexiveModule 2013-03-22 19:22:38 2013-03-22 19:22:38 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 16D90 msc 16D80 torsionless reflexive