# dual module

Let $R$ be a ring and $M$ be a left http://planetmath.org/node/365$R$-module. The dual module of $M$ is the right http://planetmath.org/node/365$R$-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$(R,R)$-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 DualModule 2013-03-22 16:00:26 2013-03-22 16:00:26 Mathprof (13753) Mathprof (13753) 10 Mathprof (13753) Definition msc 16-00 Unimodular linear functional