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∈M. Then for any f∈M*, the mapping
is a left R-module homomorphism from M* to R. In other words, the mapping is an element of M**. We call this mapping m^, since it only depends on m. For any m∈M, the mapping
is a then a right R-module homomorphism from M to M**. Let us call it θ.
Definition. Let R, M, and θ be given as above. If θ is injective, we say that M is torsionless. If θ 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