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 semi-reflexive.

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.