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 structureMathworldPlanetmath of a left module over R. The dual of that, M**, is in turn a right R-module. Fix any mM. Then for any fM*, the mapping


is a left R-module homomorphismMathworldPlanetmath 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 mM, 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 injectivePlanetmathPlanetmath, we say that M is torsionless. If θ is in addition an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, we say that M is reflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath. A torsionless module is sometimes referred to as being semi-reflexive.

An obvious example of a reflexive module is any vector spaceMathworldPlanetmath over a field (similarly, a right vector space over a division ring).

Some of the properties of torsionless and reflexive modules are

Title reflexive module
Canonical name ReflexiveModule
Date of creation 2013-03-22 19:22:38
Last modified on 2013-03-22 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