# 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 |