dual module

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

It is denoted by $M^{\ast }$. The elements of $M^{\ast }$ are called linear functionals.

The action of $R$ on $M^{\ast }$ is given by $(fr)(m)=(f(m))r$ for $f\in M^{\ast }$, $m\in M$, and $r\in R$.

If $R$ is commutative, then every $M$ is an http://planetmath.org/node/987$\mathrm{(}\mathrm{R}\mathrm{,}\mathrm{R}\mathrm{)}$-bimodule with $rm=mr$ for all $r\in R$ and $m\in M$. Hence, it makes sense to ask whether $M$ and $M^{\ast }$ are isomorphic. Suppose that $b:M\times M\to R$ is a bilinear form. Then it is easy to check that for a fixed $m\in M$, the function $b(m,-):M\to R$ is a module homomorphism, so is an element of $M^{\ast }$. Then we have a module homomorphism from $M$ to $M^{\ast }$ given by $m\mapsto b(m,-)$.

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