dual module

Let R be a ring and M be a left http://planetmath.org/node/365R-module. The dual module of M is the right http://planetmath.org/node/365R-module consisting of all module homomorphisms from M into R.

It is denoted by M. The elements of M are called linear functionals.

The action of R on M is given by (fr)(m)=(f(m))r for fM, mM, and rR.

If R is commutativePlanetmathPlanetmathPlanetmath, then every M is an http://planetmath.org/node/987(R,R)-bimodule with rm=mr for all rR and mM. Hence, it makes sense to ask whether M and M are isomorphicPlanetmathPlanetmathPlanetmath. Suppose that b:M×MR is a bilinear formPlanetmathPlanetmath. Then it is easy to check that for a fixed mM, the function b(m,-):MR is a module homomorphism, so is an element of M. Then we have a module homomorphism from M to M given by mb(m,-).

Title dual module
Canonical name DualModule
Date of creation 2013-03-22 16:00:26
Last modified on 2013-03-22 16:00:26
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 10
Author Mathprof (13753)
Entry type Definition
Classification msc 16-00
Related topic Unimodular
Defines linear functional